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PRACTICAL ENGINEERING DRAWING 



AND 



THIRD ANGLE PROJECTION 



F. N. \VlLLSON 



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PRACTICAL ENGINEERING DRAWING 



AND 



THIRD ANGLE PROJECTION 



FOR STUDENTS IN SCIENTIFIC, TECHNICAL, AND MANUAL TRAINING SCHOOLS 

AND FOR 
ENGINEERING AND ARCHITECTURAL DRAUGHTSMEN, SHEET METAL WORKERS, Etc. 



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Fredefiek fleoaton Willson, C.E., fl.JVI., 



Professor of Descriptive GeoJ'tetry^ Stercoiomy and TeclLniccil Drawing 

ill the 

John C, Green School of Seience^ Princeton University. 



PUBLISHED BY THE AUTHOR 
PRINCETON, N. J. 
1897 
L. 



2181 



ALTHOUGH the chapters here presented are taken from Theoretical and Practical Giriphics — the 
author's more extended treatise on the theory and applications of descrij^tive geometry and 
mechanical drawing, they were iirejjared with a view to their separate issue in this form, and are 
independent of the other matter with whicli they are paged. 



COPvniGHTED, Ifl'je, BY FRED'K n. WILLSON. 






TABLE OF CONTENTS 



NOTE-TAKING, DIMENSIONING, ETC. 
Technical B'ree-Hand Sketching and Lettering. — 
Note -Taking from Measurement. —Dimension- 
ing. — Conventional Representations. 

Pages 5 - lo. 

THE draughtsman's EQUIPMENT. 
The Choice and Use of Drawing Instruments and 
the Various Elements of the Draughtsman's 
Equipment. — General remarks preliminary to 
instrumental work. 

Pages II -20. 

EXERCISES FOR PEN AND COMPASS. 
Kinds and Signification of Lines. — Designs for 
Elementary Practice with the Right Line Pen. — 
Standard Methods of Representing Materials. — 
Line Shading. — Plane Problems of the Right 
Line and "Circle, including Rankine's and 
Kochansky's approximations. — Exercises for the 
Compass and Bow -pen, including uniform and 
tapered cur\'es. — The Anchor Ring. — The Hy- 
perboloid. — A Standard Rail Section. 

Pages 21-38. 

ON HIGHER PLANE CURVES AND THE HELIX. 

Regarding the Irregular Curve. — The Helix. — 
The Ellipse, Hyperbola and Parabola, by various 
methods of construction. — Homological Plane 
Cur\"es, — Relief - Perspective. — Link - Motion 
Curves. — Centroids. — The Cycloid. — The 
Companion to the Cycloid. — The Curtate and 
Prolate Trochoids, — Hypo - , Epi - , and Peri- 
Trochoids. — Special Trochoids, as the Ellipse, 
Straight Line, Limaijon. Cardioid, Trisectrix. 



Involute and Spiral of Archimedes. — Parallel 
Curves, — Conchoid. — Quadratrix. — Cissoid. 

— Tractrix. — Witch of Agnesi. — Cartesian 
Ovals. — Cassian Ovals. — Catenary. — Logarith- 
mic Spiral. — Hyperbolic Spiral. — Lituus. — 
Ionic Volute. 

Pages 39-78. 

TINTING AND SHADING. 
Brush Tinting, Flat and Graduated. — Masonry, 
Tiling, Wood Graining, River- Beds, etc., with 
brush alone, or in combined brush and line work. 
Pages 79-87. 

THE LETTERING OF DRAWINGS. 
Free- Hand Lettering. — Mechanical Expedients. 

— Proportioning of Titles. — Discussion of 
Forms. — Half- Block, Full Block and Railroad 
Types. — Borders and how to draw them. 
(Alphabets in Appendix). 

Pages 88-96. 

BLUE-PRINTING AND OTHER PROCESSES. 
The Blue -print Process. — Photo-, and other Re- 
productive Graphic Processes, including Wood 
Engraving, Cerography, Lithography. Photo- 
lithography, Chromo-lithography, Photo-engrav- 
ing, "Half-Tones," Photo-gravure and allied 
processes.— How to Prepare Drawings for Illustra- 
tion. 

Pages 97- 103. 

THIRD ANGLE PROJECTION. — WORKING 
DRAWINGS. 
Projections and intersections by the Third Angle 
Method. — The Development of Surfaces, for 



Sheet Metal or Arch Constructions. — Working 
Drawings of Bridge Post Connection, — Struc- 
tural Iron. — Spur Gearing (Approximate Invo- 
lute Outlines). — Helical Springs, Rectangular 
and Circular Section. — Screws and Bolts (U. S. 
Standard), and Table of Proportions. 

Pages 131- 180. 

AXONOMETRIC (INCLUDING 'SOMETRIC) PRO- 
JECTION. —ONE - PLANE DESCRIPTIVE 
GEOMETRY. 

Orthographic Projection upon a Single Plane. — 
Axonometric Projection. — General Fundamental 
Problem, inclinations known for two of the three 
axes.' Isometric Projection vs. Isometric Draw- 
ing. — Shadows on Isometric Drawings. — Tim- 
ber Framings and Arch Voussoirs in Isometric 
View. — One-Plane Descriptive Geometry. 

Pages 241 - 247. 

OBLIQUE PROJECTION, 

Oblique or Clinographic Projection, Cavalier Per- 
spective, Cabinet Projection, Military Perspec- 
tive. — Applications to Timber Framings, Arch 
Voussoirs and Drawing of Crystals. 

Pages 248- 250. 

APPENDIX. 

Table of the Proportions of Washers. — Working 
Drawings of Standard 100 - lb. Rail, and of 
Allen - Richardson Slide Valve. — Designs for 
Variation of Problems in Chapters X, XV and 
XVI. —Alphabets. 

Pages 251 -268. 



FREE-HAND DRAWING. 



CHAPTER II. 



ARTISTIC AND TECHNICAL FEEE-HAND DRAWING.— SKETCHING FROM MEASUREMENT. — FREE- 
HAND LETTERING.— CONVENTIONAL REPRESENTATIONS. 



20. Drawings, if classified as to the method of their production, are either free-hand or mechanical; 
while as to purpose they may be working drawings, so fully dimensioned that they can be worked 
from and what they rejaresent may be manufactured; or finished drawings, illustrative or artistic in 
character and therefore shaded either with jjeu or brush, and ha^'ing no hidden jjarts indicated by 
dotted lines as in the jDreceding division. Finished drawings also lack figured dimensions. 

Working drawings of parts or " details " of a structure are called detail drawings; while the 
representation of a structure as a whole, with all its details in their prof)er relative position, hidden 
parts indicated by dotted lines, etc., is termed a general or assembly drawing. 

21. While mechanical drawing is involved in making the various essential %dews — jilans, eleva- 
tions and sections — of all engineering and architectural constructions, and in solving the f)roblems of 
form and relative position arising in their design, yet, to the engineer, the ability to sketch effectively 
and rapidly, free-hand, is of scarcely less im23ortance than to handle the drawing instruments skill- 
fully ; while the success of an architect depends in still greater measure upon it. 

We must distinguish, however, between artistic and technical free-hand work. The architect must 
be master of both; the engineer necessarily only of the latter. 

To secure the adoption of his designs the architect relies largely upon the effective way in which 
he can finish, either with pen and ink or in water- colors, the perspectives of exterior and interior 
views; and such drawings are judged mainly from the artistic standjDoint. While it is not the 
province of this treatise to instruct in such work a word of suggestion may properly be introduced 
for the student looking forward to architecture as a jDrofession. He should j^rocure Linfoot's Picture 
Making in Pen and Ink, Miller's Essenticds of Perspective and Delamotte's Art of Sketching from Nature; 
and with an experienced architect or artist, if jjossible, but otherwise by himself, master the prin- 
ciples and act on the instructions of these writers. 

22. Since the camera makes it, fortunately, no longer essential that a civil engineer should be a 
landscape artist as well, his free-hand work has become more restricted in its scope and more rigid 
in its character, and like that of the machine designer it may properly be called technical, from its 
object. Yet to attain a sufficient degree of skill in it for all practical and commercial j^urposes is 
possible to all, and among them many who could never hope to produce artistic results. It is con- 
fined mainly to the making of working sketches, conventional representations and free-hand lettering, and 
the equipment therefor consists of a pencil of medium grade as to hardness; lettering pens — Falcon 
or Gillott's 303, with Miller Bros. "Carbon" pen No. 4; either a note-book or a sketch-block or 
pad; also the following for sketching from measurement: a two -foot pocket -rule; calipers, both 
external and internal, for taking outside and inside diameters; a pair of pencil compasses for making 
an occasional circle too large to be drawn absolutely free-hand; and a steel tape-measure for large 
work, if one can have assistance in taking notes, but otlierwise a long rod graduated to eighths. 



THEORETICAL AND PRACTICAL GRAPHICS. 



23. In the evolution of a machine or other engineering project the designer places his ideas on 
paper in the form of rough and mainly free-hand sketches, beginning with a general outline, or 
"skeleton" drawing of the whole, on as large a scale as jDOSsible, then filling in the details, separate 
— and larger — drawings of which are later made to exact scale. While such preliminary sketches are 
not drawn literally " to scale " it is ob^'iousl}^ desirable that something like the relative proportions 
should be preserved and that the closer the approximation thereto the clearer the idea they will 
give to the draughtsman or workman who has to work from them. A habit of close observation 
must therefore be cultivated, of analysis of form and of relative direction and proportion, bj^ all 
who would succeed in draughting, whether as designers or merely as copyists of existing construc- 
tions. While the beginner belongs necessarily in the latter categorj' he must not forget that his aim 
should be to place himself in the ranks of the former, both by a thorough mastery of the funda- 
mental theorj^ that lies back of all correct design and hj such training of the hand as shall facilitate 
the graphic expression of his ideas. To that end he should improve every opjjortunity to put in 
practice the following instructions as to 

SKETCHING FROM MEASUREMENT, 

as each structure sketched and measured will not only give exercise to the hand but also prove a 
valuable object lesson in the proportioning of parts and the modes of their assemblage. 

A free-hand sketch maj' be as good a working drawing as the exactly scaled — and usuallj' 
inked — drawing that is generallj' made from it to be sent to the shop. 

"While several views are usually required, yet for objects of not too comijlicated form, and whose 
lines lie mainly in mutually perpendicular directions, the method of representation illustrated by Fig. 
7, is admirablj' adapted,* and ob^dates 'all necessity for additional sketches. It is an oblique -projection 




FREE-HAND SKETCH OF TIMBER FRAMING. 



(Art. 17) the theory of ^\-hose construction will be found in a subsec^uent chapter, but with regard 
to which it is sufficient at this point to say that the right angles of the front face are seen in 
their true form, while the other right angles are shown either of 30°, 60°, or 120°; although almost 
any oblique angle will give the same general effect and may be adopted. Lines parallel to each 
other on the object are also i^arallel in the drawing. 

Draw first the front face, whose angles are seen in their true form ; then run the oblique lines 
off in the direction which will give the best view. (Refer to Figs. 42, 44, 45 and 46.) 

24. While Fig. 7 gives almost the jjictorial effect of a true perspective and the object requires 
no other clescriiDtion, yet for comjjlicatecl and irregular forms it gives place to the plan - and - eleva - 
tion mode of representation, the plan being a top and the elevation a jront ^dew of the object. And 

* The figures iu this chapter are photo - reproductions of free-hand work and are intended not only to illustrate the text 
but also to set a reasonable standard for sketch -notes. 



SKETCHING FROM MEASUREMENT. 



if two views are not enough for clearness as many more should be added as seem necessary, includ- 
ing what are called .sections, which represent the object as if cut apart by a plane, separated and a 
view obtained perpendicular to the cutting plane, showing the internal arrangement and shape of parts. 

In Fig. 8 we have the same object as in Fig. 7, but represented by the method just mentioned. 
The front %'iew (elevation) is evidently the same in hoth Figs. 7 and 8, except that in the latter we 
indicate by dotted lines the hidden recess which is in full sight in Fig. 7. 

The view of the top is placed at the top in conformity to the now quite general practice as to 
location, ^dz., grouping the various sketches about the elevation, so that the view of the left end is 
at the left, of the right at the right, etc. 






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FREE- HAN O SKETCH OF TrMBER FRAMING, fN PLAN AND ELEVATION . 



In these views, which fall under Art. 19 as to theoretical construction, entire surfaces are fro- 
kcted as straight lines, as G B C H in the straight line H' C". Were this a metallic surface and 
"finished'' or "machined" to smoothness, as distinguished from the surface of a rough casting, that 
fact would be denoted by an "/" on the line H' C" which represents the entire surface, the cross- 
line of the "/" cutting the line obliquely, as shown. 

CENTEE - LINES. — DIMENSIONING. 

25. Dimensioning. In sketching, centre-lines and all important centres should he located first, 
and measurements taken from them or from finished surfaces. 

Feet and inches are abbreviated to "Ft.," and "In.," as 4 Ft. 6| In.: also written 4' 6|", and 
occasionally 4 Ft. 6f". A dimension sh(juld not be written as an improper fraction, i^" for ex- 



ample, but as a mixed number, 1| 



Fractions should have horizontal dividing lines. 



Not only should dimensions of successive parts be given but an "over -all" dimension, which, it 
need hardly be said, should sustain the axiom regarding the whole and the sum of its parts. 

Dimensions sliould read in line with the line they are on, and either from the bottom or the 
right hand. 

The arrow tijis should toucli the lines between which a distance is given. 



THEORETICAL AND PRACTICAL GRAPHICS. 



Extension lines should be drawn and the dimension given outside the drawing whenever such- 
course will add to the clearness. (See D' F', Fig. 8.) 

An opening should always be left in the dimension line for the figures. 

In case of very small dimensions the arrow tips may be located outside the lines, as in Fig. 9, 
and the dimension indicated by an arrow, as at A, or inserted as at B if there is room. 

Should a piece of uniform cross-section (as, for example, a rail, angle -iron, channel bar, Phoenix 
column or other form of structural iron) be too long to be represented in its proper relative length 
on the sketch it may be broken as in Fig. 9, and the form of the section (which in the case sup- 
posed will be the same as an end view) may be inserted with its dimensions, as in the shaded 
figure. If the kind of bar and the number of f)0unds per j^ard are known the dimensions can be 
obtained by reference to the handbook issued by the manufacturers. 



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FREE-HAND . SKETCH OF A CHANNEL BAR. 



The same dimension should not appear on each view, but each dimension must be given at least 
once on some view. 

Notes on Riveted Work, Pins, Bolts, Screws and Nuts. In riveted work the " pitch " of the rivets, 
i. e., their distance from centre to centre (" c. to c") should lie noted, as also that between centre 
lines or rows, and of the latter from main centre lines. Similarly for bolts and holes. If the latter 
are located in a circle note the diameter of the circle containing their centres. Note that a hole 
for a rivet is usually about one -half the diameter of the forged head. 

In measuring nuts take the width between parallel sides (" width across the flats ") and abbreviate 
for the shape, as "sq.," "hex.," "oct." 

For a piece of cylindrical shape a frequently used symbol is the circle, as 4" O (read "four 
inches, round," not around,) for 4" diameter; but it is even clearer to use the abbreviation of the 
latter word, viz., " diam." 

In taking notes on bolts and screws the outside diameter is sufficient if they are " standard," 
that is, proportioned after either the Sellers (U. S. Standard) or Whitworth (English Standard) sys- 
tems, as the prop)ortions of heads and nuts, number of threads to the inch, etc., can be obtained 
from the tables in the Appendix. If not " standard " note the number of threads to the inch. Record 
whether a screw is right- or left-handed. If right-handed it will advance if turned clock -wise. The 
shape of thread, whether triangular or square, would also be noted. 

Notes on Gearing. On cog, or " gear," wheels obtain the distance between centres and the number 
of teeth on each wheel. The remaining data are then obtained by calculation. 

Bridge Notes. In taking bridge notes there would be required general sketches of front and end 
view; of the flooring system, showing arrangement of tracks, ties, guard -beam and side -walk; a 
cross -section; also detail drawings of the top and foot of each post- connection in one longitudinal 
line from one end to the middle of the structure. In case of a double -track bridge the outside 
rows of posts are alike but differ from those of the middle truss. 



CUXVK.XTIoyAL REJ> UESEXTA T 1 X S. — F R E E- H A M) LETTERING. 9 

All notes should Ije taken (in as lar}>-e a seale as possil-ile, and so indexed that drawings of 
parts ma)^ readily be understood in their relation to the whole. 

The foregoing hints might be considerably extended to embrace other and special cases, but 
experience will prove a sufficient teacher if the student will act on the suggestions given, and will 
remember that to get an excess of data is to err on the side of safety. It need hardly be added 
that what lias preceded is intended to be merely a partial sunnnary of the instructions which would 
be o-iven in the more or less brief practice in technical sketching which, presumably, constitutes a 
part of every course in Graphics; and that unless the draughtsman can be under the direction of 
a teacher he will be alile to sketch much more intelligently after studying more (jf the theory 
involved in Mechanical Drawing and given in the later pages of this work. 

COXVENTIOX.VL REPRESENTATIONS. 

26. Tonventional representations of the natural features of the country or of the materials of 
construction are so called on the assumption, none too well founded, that the engineering profession 




has agreed in convention that they shall indicate that which they also more or less resemble. While 
there is no universal agreement in this matter there is usually but little ambiguity in their use, 
especially in those that are drawn ft-ee-hand, since in them there can be a nearer approach to the 
natural appearance. This is well illustrated by Figs. 10 and 11. 



ki. 





In addition to a rock section Fig. 11 ("a) shows the method of indicating a mud or sand bed 
witli small random lioulders. 

Water either in section or as a receding surface may be shcjwn l\v parallel lines, the spaces 
between them increasing gradually. 

Conventional representations of wood, masonry and the metals will be found in Chapter VI, after 
hints on coloring have lieen given, the foregoing figures ajipearing at this point merely to illustrate, 
in black ami wliite, one of the important divisions of technical free-hand work. Those, however, who 
have already had some practice in drawing may undertake them either with pen and ink or in 
colors, in the latter case observing the in.structions of Arts. 237-241 for wood, while for the river 



10 THE B E riCA L A ND PEA C TIC A L G B A PHIC S. 

sections they may employ burnt loiiber undertone for the earthy bed, pale blue or india ink tint for 
the rock, and pni>i>iiaH blue for the water lines. 

FREE-HAKI) LETTERIXG. 

27. Although later on in this work an entire chapter is dev(^ted to the subject of lettering, yet 
at this point a word should be said regarding those forms of letters whieli ought to be mastered, 
early in a draughting course, as the most serviceable to the practical worker. . 

:Fig-. 3.2. 

ABCDEFGHIJKLMNOPQRSTUVWXYZ& 

1234567890 

ABCDEFGHIJKLMNOPQRSTUVWXYZdL 

I234567890 

The first, known as the Gothic, is the simplest form of letter, and is illustrated in both its 

vertical and inclined (or Italic) forms in Fig. 12. It is much used in dimensioning, as well as for 

12 (a.) titles. The lettering and numerals are Gothic in Figs. 7 and 8, with the exception 

^. of the 1 and 4, which, by the addition of feet, are no longer a pure form although 

O^V^^M^ T' enhanced in appearance. 

In Fig. 12 (a) some modifications of the forms of certain numerals are shown; 
also the omission of the dividing line in a mixed number, as is customary in some offices. 

For Gothic forms a coarse j^en is necessary, and shading is to be avoided, the object being to 
get all lines of uniform weight. Miller Bros'. "Carbon" pen. No. 4, is well adapted to them for w^ork 
on a fairly smooth surface and with a free -flowing ink. 

Fig. 13 illustrates the Italic (or inclined) form of a letter which Avhen vertical is known as the 
Roman. The Roman and Italic Boman are much used on Government and other map work, and in 

^ig". 13- 

^ B C D I] F G H I J K L M W P Q E S 

12 3 4 5 TUT W X T Z 67890 

a~b c d e f CI Iviyl k I m n ojj q^r s tTi v w x ij z 

the draughting offices of many j)rominent mechanical engineers. Regarding them the student may 
profitably read Arts. 260-262. Make the spaces between letters as nearly uniform as possible, and 
the small letters usually about three - fifths the height of the capitals in the same line. 

For Roman and other forms of letter requiring shading use a fine pen; Gillott's No. 303 for 
small work, and a " Falcon " pen for larger. 

A form of letter much used in Europe and growing in favor here is the Soennecien Roimd 
Writing, referred to more particularly in Art. 265 and illustrated by a complete alphabet in the 
Appendix. The text -book and special pens required for it can be ordered through any dealer in 
draughtsmen's supplies. 



THE DRAUGHTS MAX'S EQUIPMENT. 



11 



CHAPTER III. 



DRAWING INSTRUMENTS AND MATERIALS. — INSTRUCTIONS AS TO USE. 

AND TECHNICALITIES. 



-GENERAL PRELIMINARIES 



Fig-- IS. 




S 



4 



28. The drauji'ht.snuiu s equipment for graphical work .should Ije the l.iest con- '^^s- 1-*- :Fig-. is. 
sistent with his means. It is mistaken economy to buy inferior instruments. 
The best obtainalile -will be found in the end to have Ijeen the cheapest. 

The set of instruments illustrated in the following figures contains only those 
which may lie considered a))solutely essential for the beginner. 

THE DU.VWINC PEN. 

The right line pen (Fig. 14) is onlinarilj- used for drawing .straight lines, 
with either a rule or triangle to guide it ; liut it is also employed for the draw- 
ing of curves when directed in its motion l>y curves of wood or liard rubber. 
For average work a pen alxiut five inches long is best. 

The figure illustrates the most ajiproved type, i. e., made from a single piece 
of steel. The distance between its ])oints, or "nibs," is adjustable l>y means of 
the screw H. An older form of pen has the outer blade connected with the 
inner by a liinge. The convenience ^\\ih. which such a pen may be cleaned is 
more than offset liy the certainty that it will not do satisfactory work after the 
joint has become in the slightest degree loose and inaccurate through wear. 

29. If the points wear unerj^ually or become blunt the draughtsman may 
sharpen them readily himself ujion a fine oil-stone. Tlic ])roce,ss is as follows: 

Screw up the blades till they nearly touch. Incline the pen at a small angle 
to the surface of the stone and draw it lightly fi-om left to right (sujiposing 
the initial position as in Fig. Ifi). Before reaching the right 
enrl of the stone liegin turning the pen in a plane perpendic- 
ular to the surface, and draw in the opposite direction at the 
same angle. Alter frequent examination and trial, to see that 
the blades liave l)ecomc equal in length and similarly rounded, the jDrocess is 
completed by lightly dressing the outside of eacli blade separately upon the 
stone. Xo griniling shoulil l)e done on the inside of the blade. Any "burr" 
or rough cilge resulting from tlie operation may lie removed mth fine emery 
paper. For tlie liest results, obtained in the shortest pos.sible time, a magnifying glass should be- 
hould take particular notice c^f the shape of the pen when new, as a standard 



Tile student 
aimed at when compelled to act on the above suggestions. 



used, 
to be 

'■V). The ijcn may be supplied with ink by means of an ordinary writing pen dipped in the ink 
and then passed between the blades; or by u.sing in the same manner a strip of Bristol board 
about a quarti-r of an inch in width. Should any fresh ink get on the outside of the pen it must 



12 



THEORETICAL AND PR A C T T CA L G R A PHI C S. 



be removed ; otherwise it will be transferred to the edge of the rule mikI thence to the jjuper, caus- 
ing a blot. 

31. As with the jjencil, so with the pen, horizontal lines are to be dra\\'n from Itfi to right, 
while vertical or inclined lines are drawn either from or toward the worker, according to tlie position 
of the guiding edge with respect to the line to be drawn. If the line 
were m n, Fig. 17, the motion would be away from the draughtsman, 
i. e., from n toward m ; while op would be drawn foivnrd the worker, 
being on the right of the triangle. 

32. To make a sharply defined, clean-cut line — the only kind 
allowable — the pen should be held lightly but firmly with one blade 
resting against tire guiding edge, and with both jjoints resting equally 
upon the paper so that they may wear at the same rate. 

83. The inclination of the pen to the paper may best be about 70°. When properly held the 
pen will make a line about a fortieth of an inch from the edge of the rule or triangle, leaving 
visible a white line of the paper of that width. If, then, we wish to connect two points by an 
inked straight line, the rule must be so placed that its edge will Ije from them the distance indicated. 

It need hardly be said that a drawing -ijen should not be pushed. 

The niore frequently the draughtsman will take the trouble to clean out the point t)f the pen 
and supjDly fresh ink the more satisfactory results will he obtain. When through with the pen clean 
it carefully, and lay it away with the points not in contact. Equal care slmuld lie taken of all the 
instruments, and for cleaning them nothing is superior to chamois skin. 




DIVIDERS. 

34. The hair-sjiring di\dders (Fig. 15) are employed in dividing lines and spacing off distances, 
and are capiable of the most delicate adjustment by means of the screw G and si:)ring in one of the 
legs. When but one j)air of dividers is purchased the kind illustrated should have the preference 
over i^lain dividers, which lack the spring. It will, however, be frequently found convenient to have 
at hand a jDair of each. Should the joint at F become loose through wear it can be tightened by 
means of a key having two projections which fit into the holes shown in the joint. 

35. In spacing oif distances the jDressure exerted should be the slightest consistent with the loca- 
tion of a point, the jDuncture to be merely in the surface of the paper and the f)oints determined 

by lightly pencilled circles about them, thus q q . In laying off several ecjual distances 

along a line all the arcs described by one ^igr- is- jgg of the dividers should be on the 

same side of the line. Thus, in Fig. 19, with b the first centre of turning, the leg x describes the 



X'ig'- IS. 




M a 



_/., w 



arc R, then rests and pivots on c while the leg y describes the arc S; x then traces arc T, etc. 



THE COMPASSES.— HOW-ri'JXCIL .1X1) PEN. 



13 



COMPASS SET. 
^ig-- SO- S'igr- 21- E'ig-- 22. 



.»J 



36. The compasses (Fig. 20) rcseiuMe the dividers in tnrm and nuiij be used to jjerform the same 
office, but are usually employed for tlic drawing of circles. Unlike the dividers one or both of the 

legs of compasses are detachable. Those illustrated have one perma- 
nent leg, with pivot or "needle-point'" adjustable by means of screw R. 
The other leg is detachable by turning the screw 0, Avhen the pen leg 
LM (Fig. 21) may be inserted for ink work; or, where large work is 
involved, the lengthening bar on the right (Fig. 22) may be first 
attached at and the pencil or pen leg tlien inserted at /. The 
metallic point held by screw N is usually rejilaceil liy a liard lead, 
sharpened as indicated in Art 54. 

37. When in use the legs should be bent at the joints I' and L, 
so that they will lie perjiendicular to the paper when the compasses are 
held in a vertical plane. The turning may be in either direction, but 
is usually " clock- wise ; " and the cunipasses may be slightly inclined 
toward the direction of turning. A\'hen so used, and if no undue 
pressure be exerted on the pivot leg, there should be but the slightest 
puncture at the centre, while the pen points having rested equally upon 
the paper have sustained ecpial wear, and the resulting line has been 
sharply defined on both sides. Obviously the legs must be re -adjusted 
as to angle, for any material change in the size of the circles wanted. 

The compasses shuuld be held and turned l)y the milled head 
which 131'ojects above the joint i\'. 

Dividers and compasses should o]ien an<l shut with an absolutely 
uniform motion and somewhat stifflv. 



k 




BOW' - PEXCIL AND PEN. 

38. For extremely accurate work, 
in diameters fi-om one -sixteenth of an inch to about 
two inches, the hoxv -pencil (Fig. 23) and bow -pen (Fig. 
24) are especially adapted. The pencil -Ixiw has a 
needle-point, adjustable by means of screw £, which 
gives it a great advantage over the fixed pivot -point 
of the bo-w-pen, not alone in that it permits oi more 
delicate adjustment for unusualty small work but also 
because it can be easily rei^laced by a new one in 
case of damage ; whereas an injury to the other ren- 
ders the whole instrument useless. For verj^ small 
circles the needle-point should project very slightly beyond the pen- 
point ; theoretically by only the extremely small distance tlie needle- 
point is expected to sink into the i^aper. 

The sirring of either bow should be strong; otherwise an attemi)t 
at a circle will result in a si^iral. 

It will save wear upon the threads of the milled heads A and C 
if the draughtsman will press the legs of the bow together with his 
left hand and run the head up loosely on the screw with his right. 



E-i^. 23. 



^'ig, 2-5:. 





14 THEORETICAL AXD PRACTICAL GRAPHICS. 

39. To the above described — which we may call the ininimum set of instruments — might be 
advantageously added a pair of bow -spacers (small dividers shaped like Fig. 24); beam -compasses, 
for extra large circles; parallel - rule ; proportional dividers, and an extra — and larger ^ — right -line pen. 

40. The remainder of the iHressari/ equipment consists of j^aper; a drawing-board; T-rule; tri- 
angles or "set squares;" scales; pencils; India ink; water colors; saucers for mixing ink or colors; 
brushes; water-glass and sponge; irregular Cor "French") curves; India rubber; erasing knife; pro- 
tractor; file for sharjiening jaencils, or a pad of fine emery or sand i^aper; tlunnb -tacks (or "drawing- 
pins"); horn centre, for making a large number of concentric circles. 

PAPER AND TRACING CLOTH. 

41. Drawing ijapcr may lie purchased hy the sheet or roll and either unmounted or mounted, 
i. e., "backed" by muslin or heavy card-board. Smooth or "hot-pressed" paper is best for drawings 
in line -work only; but the rougher surfaced, or " cold- pressed," should always be employed when 
brush-work in ink or colors is involved: in the latter case, also, either mounted paper should be 
used or the sheets " stretched " l_\v the process described in Art. 44. 

42. The names and sizes of sheets are: — 
Cap 13 X 17 Elephant 23 x 28 

Demi 15 x 20 Atlas 26 X 34 

Medium 17 X 22 Columljia 23 x 35 

Royal 19 x 24 Double Elephant 27 x 40 

Super Royal 19 X 27 Antiijuarian 31 x 53 

Imperial 22 x 30 

43. There are many makes of first-class papers, but the best known and still proliably the most 
used is Whatman's. The draughtsman's choice of pajjer must, howe'S'er, be determined largely by the 
value of the dra'(\'ing to lie made upon it, and lij' the prolialile usage to which it will be subjected. 

Where several copies of one drawing were desired it has been a general practice to make the 
original, or " construction " drawing, with the pencil, on paper of medium grade, then to lay over 
it a sheet of tracing -doth and copy ujion it, in ink, the lines underneath. Upon placing the tracing 
cloth over a sheet of sensitized paper, exjiosing lioth to the light and then immersing the sensitive 
pajjer in water, a copy or print of the drawing was found upon the sheet, in white lines on a blue 
ground — the well-kn(n\-n blue -print. The time of the draughtsman ma_V, however, be economized, as 
also his purse, by making the original drawing in ink upon Crane's Bond paper, which combines in 
a remarkable degree the qualities of transparency and toughness. About as clear liluc- prints can be 
made with it as with tracing - cloth, j'et it will stand se^•ere usage in the shop or the drafting -room. 

Better paj^ers may yet be manufactured for such purposes, and the progressive draughtsman will 
be (3n the alert to avail himself of these as of all genuine improvements Ufion the materials and 
instruments before employed. 

44. To stretch paper tightly upon tlie board lay the sheet right side up.* place the long rule 
with its edge about one -half inch back from each edge of the paper in turn, and fold up against it 
a margin of that width. Then thoroughly dampen the back of the paper with a full sponge, except 
on the folded margins. Turning the paper again face uji gum the margins with strong mucilage or 
glue, and quickly Itut firmly press opposite edges down simultaneously, long sides first, exerting at the 
same time a slight outward pressure with the hands to bring the paper down somewhat closer to 



*The "right side" of a sheet is, presumably, that towiird one when — on holding it up to the light — the manufiictiirer'B. 
name, in water- mark, reads correctly. 



TEA crxa-CL () ril. — DRA WfXG B l)ARD.— T-R F L K.— TR lA X G L E S. 15 

the board. Until the gum ''set:^," so that the paper adheres i)erteetly where it should, the latter 
should not shrink ; hence the necessity for so completely soaking it at first. The sponge may be 
applied to the fair of the paper jirovided it is not rubbed over the surface, so as to damage it. 
The stretch should be ln)rizontal when drying, and no excess of water should be left standing on 
the surface; otherwise a water -mark will form at the edge of each pool. 

45. When tracing- cfoi/t is used it must !>e fastened smoothly, with thumb-tacks, over the drawing 
to be copied, and the ink lining done u])on the glazed side, any l)rush work that may be recjuired — 
either in ink or colors — lieing always done u]ion the dull side of the cloth after the outlining has 
been completed. 

If the glazed surface be first dusted with powdered j)i]je-clay applied with chamois skin it will 
take the ink much more readily. 

AMien erasure is necessary use the rublier. after which the surface may be restored for further 
pen -work \>y rubbing it with soapstone. 

Tracing -cloth, like drawing paper, is most convenient to work u])on if perfectly flat. When either 
has been puix-hased by the roll it should therefore be cut in sheets and laid away for some time in 
drawers to become flat before needed for use. 

I)HAWI.\(; BOARD. 

46. The drawing board should l>e slightly larger than the pajjer for which it is designed and 
of the most thoroughly seasoned material, jjreferably some ^(ift wood, as pine, to facilitate the use of 
the drawing-pins or thumli -tacks. To ])revent warping it should have 1)attens of hard wood dove- 
tailed into it across the back, transversely to its length. The back of the board should be grooved 
longitudinally to a depth equal to half the thickness of the wood, Avhich weakens the board trans- 
versely and to that degree facilitates the stifl'ening action of the battens. 

For work of moderate size, on stretched j>a23er, yet without the use of mucilage, the " 2:)anel " 
board is recommended, provided that both frame and jjanel are made of the best seasoned hard wood. 

It will be found convenient for each student in a technical school to possess two boards, one 
20" X 28" for paper of Super Royal size, which is suitable for much of a beginner's work, and another 
28" X 41" for Double Elephant sheets (abfiut twice Super Rf)yal size) which are well adapted to large 
drawings of machinery, bridges, etc. A large board may of course be used for small sheets, and the 
expense of getting a second board avoided ; but it is often a great convenience to have a medium- 
sized board, especially in case the student desires to do some work outside the draughting-room. 

THE T-KULE. 

47. The T-rule should be slightly shorter than the drawing board. Its head and blade must 
have absolutely straight edges, and be so rigidly combined as to admit of no lateral play of the 
latter in the former. The head should also be so fastened to the blade as to be level with the surface 
of the board. This i^ermits the triangles to slide freely over the head, a great convenience when 
the Hnes of the drawing run close to the edge of the j^aper. (See Fig. 32.) 

The head of the T-rule should always be used along the left-hand edge of the drawing board. 

TRIANCJLES. 

48. Triangles, or " set - squares " as they are also called, can be .obtained in various materials, as 
hard rubber, celluhjid, pear -wood, mahogany and steel; and either solid (Fig. 25) or oi^en (Fig. 26). 
The open triangles are preferable, and two are required, one with acute angles of 30° and 60°, the 
other with 45° angles. Hard rubber has an advantage over metal or wood, the latter being likely 
to warp and the fomier to rust, unless plated. Celluloid is transparent and the most cleanly of all. 



16 



THEORETICAL AND PRACTICAL GRAPHICS. 



The most frequently .recurring problems involving the use of the triangles are the following: — 



ng-- as- 





49. To draw parnlld lines place either of the edges E'igr- ss. 
against another triangle or the T-rule. If then moved 
along, in either direction, each of the other edges will 
take a series of parallel positions. 

50. To draw a line perpendicular to a given line 
place the hypothenuse of the triangle, o a, (Fig. 26), 
so as to coincide with or be parallel to the given 

line; then a rule or another triangle against the base. By then turning the triangle so that the 

other side, o c, of its right angle shall be against the rule, as at o^c,, the hypothenuse will be found 

perpendicular to its first position and therefore to the given line. 

iFigr- 27. 51_ 3}) construct regular hexagons place the shortest side of the 60° 

triangle against the rule (Fig. 27) if two sides are to be horizontal, as /e 
and b c of hexagon H For vertical sides, as in H ', the position of the 
triangle is evident. By making « b indefinite at first, and knowing be — 
the length of a side, we may obtain a by an arc, centre b, radius b c. 
If the inscribed circles were given, the hexagons might also be obtained 

by drawing a series of tangents to the circles, with the rule and triangles in the positions indicated. 




THE SCALE. 

52. Bu-t rarely can a drawing be made of the same size as the object, or " full-size," as it is 
called; the lines of the drawing, therefore, usually bear a certain ratio to 'those of the object. This 
ratio is called the scale and should invariably be indicated. 

If six inches on the drawing represent one foot on the object the scale is one -half and might be 
variously indicated, thus: SCALE i; SCALE 1:2; SCALE 6 In. = 1 FT. SCALE 6" = 1' . 

At one foot to the inch any line of the drawing would be one -twelfth the actual size, and the 
fact indicated in either of the ways just illustrated. 

Although it is a simple matter for the draughtsman to make a scale for himself for any par- 
ticular case yet scales can be purchased in great variety, the most serviceable of which for the usual 
range of work is of box-wood, 12" long, (or 18", if for large work) of the form illustrated by Fig. 



^ig"- 3S- 




28, and graduated 



_3_ ■ i 
16- 8 



li: 



inches to the foot. This 



is known as the arcliitecih scale in contradistinction to the engineer's, which is 
decimally graduated. It will, however, be frequently convenient to have at 
\\\\W\A\\v\\\4 hand the latter as well as the former. 
When in use it should he laid along the line to be spaced, and a light dot made upon the 



X 11 10 9 8 76 5432 1 


b D 


I 1 1 1 1 1 1 1 1 1 1 j 






1 1 1 1 1 1 1 1 1 1 1 1 












1 1 1 1 1 1 1 1 1 1 






/ 1 1 ■ / 1 1 1 1 






1 1 1 1 1 1 1 1 1 1 I si 


X 




/// 1 1 1 1 t 1 1 






1 1 1 1 1 1 1 1 1 1 






1 1 1 1 1 1 1 1 






1 Ic Id k If Ig Ih li h Ik ll I'll 






Sill 1ft 


B 


a 







paper with the pencil, opposite the proper division on the graduated edge. A distance should rarely 



SCALES. — PENCILS. — INKS. 17 

be transferred from the scale to the drawing by the dividers, as such procedure damages the scale 
if not the paper. 

53. For special cases diagonal scales can readily be constructed. If, for example, a scale of 
3 inches to the foot is needed and measuring to fortieths of inches, draw eleven equidistant, parallel 
lines, enclosing ten equal spaces, as in Fig. 29, and from the end A lay off A B, B C, etc., each 
3 inches and representing a foot. Tlien twelve parallel diagonal lines in the first space intercept 
quarter -inch spaces on AB or ab, each representative of an inch. There being ten equal siDaces 
between B and b, the distance s x, of the diagonal 6 m from the vertical b B, taken on any horizontal 
line s X, is as many tenths of the space in B as there are sjDaces between s x and b ; six, in this case. 
The principle of construction may be generalized as follows : — 

The distance apart of the vertical lines rejaresents the units of the scale, whether inches, feet, 
rods or miles. Except for decimal graduation divide the left-hand space at top and bottom into as 
many spaces as there are units of the next lower denomination in one of the original units (feet, 
for j'ards as units ; inches in case of feet, etc.). Join the points of division by diagonal lines ; and, 
if — is the smallest fraction that the scale is designed to give, rule x + 1 equidistant horizontal lines, 
giving X equal horizontal sfjaces. The scale will then read to -th of the intermediate denomination 
of the scale. 

When a scale is properly used, the sjDaces on it which represent feet and inches are treated as 
if they were such in fact. On a scale of one -eighth actual size the edge graduated 1^ inches to the 
foot would be em2Dloyed; each 1^ inch sjsace on the scale would be read as if it were a foot; and 
ten inches, for example, would be ten of the eighth -inch spaces, each of which is to represent an 
inch of the original line being scaled. The usual error of beginners would be to divide each original 
dimension by eight and lay off the result, actual size. Tlie former method is the more expeditious. 

THE PENCILS. 

54. For construction lines afterward to be inked the pencils should be of hard lead, grade 6H 
if Fabers or VVH if Dixon's. The pencilling should be light. It is easy to make a groove in the 
paper by exerting too great pressure when using a hard lead. The hexagonal form of pencil is 
usually indicative of the finest quality, and has an advantage over the cylindrical in not rolling off 
when on a board that is slightly inclined. 

Somewhat softer pencils should be used for drawings afterward to be traced, and for the prelim- 
inary free-hand sketches from which exact drawings are to be made; also in free-hand lettering. 

Sharpen to a chisel edge for work along the edges of the T-rule or triangles, but use another 
pencil with coned point for marking off distances with a scale, locating centres and other isolated 
points, and for free-hand lettering; also sharpen the comjDass leads to a point. Use the knife for 
cutting the wood of the pencil, beginning at least an inch from the end. Leave the lead exposed 
for a quarter of an inch and shape it as desired, either with a knive or on a fine file, or a laad of 
emery paper. 

THE INK. 

55. Although for many purposes some of the liquid drawing -inks now in the market, partic- 
ularly Higgins', answer admirably, yet for the best results, either Avith pen or brush, the draughtsman 
should mix the ink himself with a stick of India — or, more correctly, China ink, selecting one of 
the higher -priced cakes, of rectangular cross -section. The best will show a lustrous, almost iridescent 
fracture, and will have a smooth, as contrasted with a gritty feel when tested by rubbing the moist- 
ened finger on the end of the cake. 



18 THEORETICAL AND PRACTICAL GRAPHICS. 

Sets of saucers, called "nests," designed for the mixing of ink and colors, form an essential part 
of an equipment. There are usually six in a set and so made that each answers as a cover for the 
one below it. Placing fi-om fifteen to twenty drops of water in one of these the stick of ink should 
be rubbed on the saucer with moderate pressure. 

To properly mix ink requires great patience, as with too great pressure a mixture results having 
flakes and sand -like particles of ink in it, whereas an absolutely smooth and rather thick, slow- 
flowing liquid is wanted, whose surface will reflect the face like a mirror. The fiiral test as to 
sufficiency of grinding is to draw a broad line and let it dry. It should then be a rich jet black, 
with a slight lustre. The end of the cake must be carefullj^ dried on removing it from the saucer 
to jjrevent its flaking, which it will otherwise invariably do. 

One may say, almost without qualification, and particularly when for use on tracing -cloth, the 
thicker the ink the better; but if it should require thinning, on saving it from one day to another — 
which is possible with the close-fitting saucers described — add a few drops of water, or of ox -gall if 
for use on a glazed surface. 

When the ink has once dried on the saucer no attempt should be made to work it up again 
into solution. Clean the saucer and start anew. 

WATER COLORS. 

56. The ordinary colored writing inks should never be used by the draughtsman. They lack 
the requisite " body " and are corrosive to the pen. Very good colored drawing inks are now manu- 
factured for line work, but Winsor and Newton's water colors, in the form called " moist," and in 
"half-i^ans" are the best if not the most convenient, for color work either with pen or brush. 
Those most frequently employed in engineering and architectural drawing are Prussian Blue, Carmine, 
Light Red, Burnt Sienna, Burnt Umber, Vermilion, Gamboge, Yellow Ochre, Chrome Yellow, Payne's 
Gray and Sepia. For some of their special uses see Art. 73. 

Although hardly properly called a color Chinese White may be mentioned at this point as a 
requisite, and obtainable of the same form and make as the colors above. 

DRAWING-PINS. 

57. Drawing-pins or thumb-tacks, for fastening paper upon the board, are of various grades, the 
best, and at present the cheapest, being made from a single disc of metal one -half inch in diameter, 
from which a section is partially cut, then bent at right angles to the surface, forming the f)oint of 
the pin. 

IRREGULAR CURVES. 

58. Irregular or French curves, also called s^veeps, for drawing non- circular arcs, are of great 
variety, and the draughtsman can hardly have too many of them. They may be either of pear 

Fig-- 30. wood or hard rubber. A thoroughly equipjDed draughting office will have a 

large stock of these curves, which may be obtained in sets, and are known 
as railroad curves, ship curves, spirals, ellipses, hyperbolas, parabolas and 
combination curves. Some very serviceable flexible curves are also in the 
market. 

If but two are obtained (which would be a minimum stock for a 
beginner) the forms shown in Fig. 30 will probably prove as serviceable as any. When employing 
them for inked work the pen should be so turned, as it advances, that its blades will maintain the 
same relation (parallelism) to the edge of the guiding curve as they ordinarily do to the edge of 




CURVES. — RUBBER. — ERASERS. — PRO TRACTORS. — BRUSHES. 



19 



the rale. And the student must content himself with drawing slightlj^ less of the curve than might 
apparently be made with one setting of the sweep, such course being safer in order to avoid too close 
an approximation to angles in what should be a smooth curve. For the same reason, when placed 
in a new position, a portion of the irregular curve must coincide with a part of that last inked. 

The pencilled curve is usually drawn free-hand, after a number of the points through which it 
should pass have been definitely located. In sketching a curve free-hand it is much more naturally 
and smoothly done if the hand is always kept on the concave side of the curve. 

INDIA EUBBER. 

59. For erasing pencil -lines and cleaning the paper india rubber is required, that known as 
"velvet" being recommended for the former purpose, and either "natural" or "sponge" rubber for 
the latter. Stale bread crumbs are equally good for cleaning the surface of the paper after the lines 
have been inked, but will damage pencilling to some extent. 

One end of the velvet rubber may well be wedge-shaped in order to erase lines without damag- 
ing others near them. 

INK ERASER. 

60. The double-edged erasing knife gives the quickest and best results when an inked line is to 
be removed. The jjoint should rarelj' be employed. The use of the knife will damage the paper 
more or less, to partially obviate which rub the surface with the thumb-nail or an ivory knife 
handle. 

PROTRACTOR. 

61. For laving out angles a graduated arc called a "protractor" is used. Various materials are 
employed in the manufacture of protractors, 
as metal, horn, celluloid, Bristol board and 
tracing paper. The two last are quite accu- 
rate enough for ordinary j)urf)0ses, although 
where the utmost precision is required, one of 
German silver should be obtained, with a 
moveable arm and vernier attachment. 

The graduation may advantageously be 
to half degrees for average work. 

To lay out an angle (say 40°) T\ath a 
protractor, the radius CH (Fig. 31) should 
be made to coincide with one side of the desu-ed angle; the centre, C, with the desired vertex; 
and a dot made with the jDencil opposite division numbered 40 on the graduated edge. The line 
MC, through this point and C, completes the construction. 

BRUSHES. 

62. Sable -hah- brushes are the best for laj'ing flat or graduated tints, with ink or colors, upon 
small surfaces; while those of camel's hair, large, with a brush at each end of the handle, are 
better adapted for tinting large surfaces. Reject any brush that does not come to a jjerfect point 
on being moistened. Five or six brushes of different sizes are needed. 

PRELIinXAEIES TO PRACTICAL WORE. 

63. The first work of a draughtsman, like most of his later j)roductions, consists of line as distin- 
guished from brush work, and for it the jjajjer may be fastened ujjon the board with thumb-tacks only. 




20 



THEORETICAL AND PRACTICAL GRAPHICS. 



There is no universal standard as to size of sheets for drawings. As a rule each draughting 
office has its own set of standard sizes, and system of preserving and indexing. The columns of 
the various engineering papers present frequent notes on these jDoints, and the best system of pre- 
serving and recording drawings, tracings and corrections is apparently in process of evolution. For 
the student the best plan is to have all drawings of the same size bound in neat but permanent 
form at the end of the course. The title-pages, which presumably have also been drawn, will suf- 
ficiently distinguish the different sets. 

In his elementary work the student may to advantage adopt two sizes of sheets which are con- 
siderably employed, 9" x 13", and its double, 13" x 18"; sizes into which a "Super Royal" sheet 
naturally divides, leaving ample margins for the mucilage in case a " stretch " is to be made. 

A "Double Elephant" sheet being twice the size of a "Super Royal" divides equally well into 
plates of the above size, biit is preferable on account of its better quality. 

To lay out four rectangles -a])on the paper locate first the centre (see Fig. 32) by intersecting 
diagonals, as at 0. These should not be drawn entirely across the sheet, but one of them will 
necessarily pass a short distance each side of the point where the centre lies — judging by the eye 
alone ; the second definitely determines ■s'lg. ss. 

the point. If the T-rule will not 
reach diagonally from corner to corner 
of the paper (and it usually will not) 
the edge may be practically extended 
by placing a triangle against but pro- 
jecting beyond it, as in the upper left- 
hand portion of the figure. 

The T-rule being placed as shown, 
with its head at the left end of the 
board — the correct and usual jDosition 
■ — draw a horizontal line X Y, through 
the centre just located. The vertical 
centre line is then to be drawn, with 
one of the triangles placed as shown 
in the figure, i. e., so that a side, as mn or tr, is perpendicular to the edge. 

It is true that as long as the edges of the board are exactly at right angles with each other 
we might use the T-rule altogether for drawing ' mutually perpendicular lines. This condition being, 
however, rarely realized for any length of time,^ it has become the custom — a safe one, as long as 
rule and triangle remain "true" — to use them as stated. 

The outer rectangles for the drawings (or "plates," in the language of the technical school) are 
completed by drawing parallels, as JN and Y N, to the centre lines, at distances from them of 9" 
and 13" respectively, laid off from the centre, 0. 

An inner rectangle, as ab c d, should be laid out on each plate, with proper margins ; usually 
at least an inch at the top, right and bottom, and an extra half inch on the left as an allowance for 
binding. These margins are indicated by x and z in the figure, as variables to which any con- 
venient values may be assigned. The broad margin x in the upper rectangle will be at the draughts- 
man's left hand if he turns the board entirely around — as would be natural and convenient — when 
ready to draw on the rectangle QY. 




mimimmmmiiimmmmfmmmmmm^^^ 



EXERCISES FOR PEN AND COMPASS. 21 



CHAPTEIt IV. 

GEADES OF LINES. — LINE TINTING. — LINE SHADING. — CONVENTIONAL SECTION-LINING.— 
FREQUENTLY RECURRING PLANE PROBLEMS.— MISCELLANEOUS PEN AND 

COMPASS EXERCISES. 

65. Several kinds of lines employed in mechanical drawdng are indicated in the figure below. 
While getting his elementary practice with the ruling -pen the student may group them as shown, 
or in any other symmetrical arrangement, either original with himself or suggested by other designs. 

Lgr- 33. 



-t- 



CENTRE LINE. If red. 



CENTRE UlNE. If black. V. 

! "^^ 

FOR ORDINhRY OUTLINES. V^ 



HIDDEl[l LINE 

I 
MEDIUM, Continuous. 



"DOTTED LI N E_ usually employed as a constructio n line. 
I 



*^ 



SHADE 



Jjt^ 



LIME 



"DOTTED LINE'I line of_motibn in Kinematic Geometry. 




When drawing on tracing cloth or tracing jjaper, for the purpose of making blue -prints, all the 
lines wiU preferably be black, and the centre and dimension lines distinguished from others as indi- 
cated above, as also by being somewhat finer than those employed for the light outlines of the 
object. Heavy, opaque, red lines may, however, be used, as they will blue -print, though faintlj-. 

There is at present no universal agreement among the members of the engineering profession as 
to standard dimension and centre Unes. Not wishing to add another to the systems already at 
variance, but preferring to facilitate the securing of the unifonuity so desirable, I have presented 
those for some time employed by the Pennsylvania Railroad and now taught at Cornell University. 

The lines of Fig. 33, as also of nearly all the other figures of this work, having been printed from blocks made by the 
cerographlc process (Art. 277), are for the most part too light to serve as examples tor machine-shop work. Fig. 80 is a sample 
of P. K. R. drawing, and is a fair model as to weight of line for working drawings. 



22 



THEORETICAL AND PRACTICAL GRAPHICS. 



A dash -and -three -dot Hue (not shown in the figure) is considerably used in Descriptive Geometry, 
either to represent an auxiliary plane or an invisible trace of any plane. (See Fig. 238). 

The so - called " dotted " line is actually composed of short dashes. Its use as a " line of 
motion" was suggested at Cornell. 

When colors are used without intent to blue -print they may l^e drawn as light, continuous lines. 
Colors will further add to the intelligibility of a drawing if employed for construction lines. Even 
if red dimension lines are used the arroio heads should invariably be black. They should be drawn 
free -hand, with a writing pen, and their points touch the lines between which they give the distance. 

66. The utmost accuracy is requisite in pencilling, as the draughtsman should be merely a copyist 
when using the pen. On a comjjlicated drawing even the kind of line should be indicated at the 
outset, so that no time will be wasted, when inking, in the making of distinctions to which thought 
has alreadj' been given during the process of construction. No unnecessary lines should l)e drawn, 
or any exceeding of the intended limit of a line if it can possibly be avoided. 

If the work is symmetrical, in whole or in part, draw centre lines first, then main outlines; and 
continue the work from large parts to small. 

The visible lines of an object are to be drawn first; afterward those to be indicated as concealed. 

All lines of the same qualitj^ may to advantage be drawn with one setting of the pen, to ensure 
uniformity; and the light outlines before the shade lines. 

In drawing arcs and their tangents ink the former first, invariably. 

All the inking may best be done at once, although for the sake of clearness, in making a large 
and complicated drawing, a portion — usuallj^ the nearest and visible parts — may be inked, the draw- 
ing cleaned, and the iDencilling of the construction lines of the remainder continued frem that j)oint. 

The inking of the centre, dimension and construction lines naturally follows the completion of 



the main design. 



















a 
















b 










1 1 2 1 3 


4 


5 


6 


7 


8 


9 


r^ 












1 


2 


3 


4 


^ 


6 


7 



















1 


2 


3 


4 


5 
















1 


2 


3 
















— ■ 








1 


2 


3 


4 


5 
















































i 
















i 






i 1 1 




1 


2 


3 


4 


^t 



67. In Fig. 34 we have a straight -line design 
usually called the " Greek Fret," and giving the 
student his first illustration of the use of the 
"shade line" to bring a drawing out "in relief" 
The law of the construction will be evident on 
examination of the numbered squares. 

Without eiatering into the theory of shadows at 

this point we may state briefly the " shoj) rule " 

for drawing shade lines, viz., right-hand and loiver. 

] That is, of any i^air of lines making the same 

turns together or representing the limit of the same 



flat surface, the right-hand line is the heavier if the pair 
is vertical, but the lower if they run horizontally; always 
subject, however, to the proviso that the line of inter- 
section of two illumin- 
ated planes is never a 
shade line. 

68. The conic section 
called the parabola fur- 
nishes another interesting exercise in i-uled lines, when 
it is represented by its tangeiits as in Fig. 35. The angle CA E 
may be assumed at pleasure, and on the finished drawing the numbers may 




SECTION-LINING. — LIJ^E-SH A DING. 



23 



^'ig-. 3S- 



be omitted, being given here merely to show the law of construction. All the divisions are equal, 
and like numbers are joined. 

Some interesting mathematical properties of the curve will be found in Chapter V. 

69. A pleasing design that will test the beginner's skill is that of Fig. 36. It is suggestive of 
a cobweb, and a skillful free-hand draughtsman could make it more realistic by adding the spider. 
Use the 60° triangle for the heavy diagonals 
and parallels to them; the T-rule for the hor- 
izontals. Pencil the diagonals first but ink them 
last. 

70. The even or flat effect of equidistant 
parallel lines is called line - tinting ; or, if repre- 
senting an object that has been cut by a plane, 
as in Fig. 37, it is called section -lining. 

The section, strictly speaking, is the part 
actually in contact with the cutting j^Iane; 
while the drawing as a whole is a sectional 
view, as it also shows what is back of the 
plane of section, the latter being always as- 
sumed to be transparent. 

Fig-. 37. Adjacent jDieces 

have the lines drawn 
in different directions 
in order to distinguish sufficiently between them. 

The curved effect on the semi - cylinder is evidently obtained Ijy prop- 
erly varying both the strength of the line and the spacing. 

71. The difference between the shading on the exterior and interior 
H of a cylinder is sharply contrasted in Fig. 38. On the concavity the 
darkest line is at the top, while on the convex surface it is near the Isot- 
tom, and below it the spaces remain unchanged while the lines diminish. 
A better effect would have been obtained in the figure had the engraver begun to increase the lines 
with the first decrease in the space between them. 

IPig-- 3S- 






The spacing of the lines, in section -lining, depends upon the scale of the drawing. It may run 
down to a thirtieth of an inch or as high as one -eighth; but from a twentieth to a twelfth of an 
inch would be best adapted to the ordinary range of work. Equal s^Dacing and not fine spacing 



24 



THEORETICAL AND PRACTICAL GRAPHICS. 



should be the object, and neither scale nor patent section-liner should be employed, but distances 
gauged by the eye alone. 

72. A refinement in execution which adds considerably to the effect is to leave a white line 
between the top and left-hand outlines of each piece and the section lines. When purposing to 
produce this effect rule light pencil lines as limits for the line -tints. 

73. If the various pieces shown in a section are of different materials there are four ways of 
denoting the difference between them : 

(a) By the use of the brush and certain water- colors, a method considerably employed in Europe, 
but not used to any great extent in this country, probably owing to the fact that it is not applic- 
able where blue -prints of the original are desired. 

The use of colors may, however, be advantageously adopted when making a highly finished, 
shaded drawing; the shading being done first, in India ink or sepia, and then overlaid with a flat 
tint of the conventional color. The colors ordinarily used for the metals are 

Payne's gray or India ink for Cast Iron. 

Gamboge " Brass (outside view). 

Carmine " Brass (in section). 

Prussian Blue - " Wrought Iron. 

Prussian Blue with a tinge of Carmine " Steel. 



X-ig-. 3S. 




Last Iran. 





StEE 



Wr't. Iran. 




PZlTlTiL. 5. 5. 

Btandai^d ^ectiens, 









Brass. 



StnnE. 








Wand. 



CappEr. 



Brick. 



CONVENTIONAL SECTION-LINING. 



25 



rig-. -41. 



More natural effects can also be given by the use of colors, in representing the other materials 
of construction; and the more of an artist the draughtsman proves to be the closer can he approx- 
imate to nature. 

Pale blue may be used for water lines; Burnt Sienna, whether grained or not, suggests wood; 
Burnt Umber is ordinarily employed for earth; either Light Red or Venetian Red are well adapted 
for brick, and a wash of India ink having a tinge of blue gives a fair suggestion of masonry; 
although the actual tint and surface of any rock can be exactly represented after a little practice 
wdth the brush and colors. These jjoints will be enlarged upon later. 

(b) By section - lining with the drawing pen in the conventional colors just mentioned, a process 
giving very handsome and thoroughly intelligible results on the original drawing, but, as before, 
unadapted to blue -printing and therefore not as often used as either of the following methods. 

(c) By section -lining uniformly in ink throughout and printing the name of the material upon 
each piece. 

^ (d) By alternating light and heavy, continuous and broken 
lines according to some law. Said "law" is, unfortunately, by 
no means universal, despite the attempt made at a recent con- 
vention of the American Societj' of Mechanical Engineers to 
secure uniformit}^ Each draughting office seems at present to 
be a law unto itself in this matter. 

74. As affording valuable examples for further exercise 
with the ruling pen the 'system of section - lines adopted by 
the Pennsylvania Railroad is presented on the opposite page. 
The wood section is an exception to the rule, being drawn 
free-hand, with a Falcon \>en. 

T'^s- -SiO. gy ^yr^y gf coutrasting free-hand with me- 

chanical work Fio;. 40 is introduced, in which 






COURSED RUBBLE MASONRY 
Light India Ink, 



RUBBLE MASONRY 
Light India Ink, 



the rings showing annual growth are drawn 





as concentric circles with the comi^ass. 

In Fig. 41 a few other sections appear, 
■0^ selected from the designs of M. N. Forney 
and F. Van Vleck, and which are fortunate arrangements. 

75. Figs. 42 and 43 are profiles or outlines of mouldings, 
such as are of frequent occurrence m architectural work. It is 
good practice to convert such views into oblique projections, giving the effect of solidity; and to 
further bring out their form by line shading. Figs. 44-46 are such representations, the front of each 
being of the same form as Fig. 42. The olalique lines are all jjarallel to each other, and ^ where 

^±S- "is. ^ig-- -43. X'ig'. 4^- 



BRICK 
Venetian Red. 



CONCRETE 
Yellow Ochre. 





^ '^ 


\ 


n ""' 






-H 




r 1" 


V 








'^ 


=-*' 


i 





'visible throughout — of the same length. Their direction should h& chosen with reference to best ex- 
hibiting the peculiar features of the object. Ob^^.ously the view in Fig. 44 is the least adapted to 
the conveying of a clear idea of the moulding, while that of Fig. 46 is evidently the best. 



26 



THEORETICAL AND PRACTICAL GRAPHICS. 



76. The student may, to advantage, design i^rofiles for mouldings and line -shade them, after 
converting them into oblique views. As hints for such work two figures are given (47-48), taken 





from actual construction in wood. By setting a moulding vertically, as in Fig. 49, and projecting 
horizontally fi-om its points, a fi-ont view is obtained, as in Fig. 50. 



Fig-. '47'. 



■FLs- -ae. 







77. The 



reverse curves 





X'ig-. SI. 






M 






n 




/'o 
















yy 








































^v^ 


/ 




m 


„¥■ 


\ 






c, 




N 



on the mouldings may be drawn with the irregular curve, (see Art 68); 
or, if composed of circular arcs to be tangent to vertical lines, by the follow- 
ing construction : — 

Let 31 and iV be the points of tangency on the verticals Mm. and N n, 
and let the arcs be tangent to each other at the middle point of the line 
MK Draw 3In and Nm perpendicular to the vertical lines. The centres, c 
and c,, of the desired arcs, are at the intersection of 31 n and Nm by per- 
pendiculars to 31 N from x and y, the middle points of the segments of 31 N. 
78. The light is to be assumed as coming in the usual direction, i. e., 
descending from left to right at such an angle that any ray would be projected on the paper at an 
angle of 45° to the horizontal. 

In Fig. 43 several rays are shown. At x, where the light strikes the cylindrical portion most 
directly— technically is normal to the surface— is "actually the brightest part. A tangent ray st gives 
t, the darkest part of the cylinder. The concave portion beginning at o would be darkest at o and 
get lighter as it apjDroaches y. 

Flat parts are either to be left white, if in the light, or have equidistant Hues if in the shade, 
unless the most elegant finish is desired, in which case both change of space and gradation of line 

must be resorted to as in Fig. 52, which represents a front view of a, 
ll^fc hexagonal nut. The front face, being parallel to the paper, receives an 
even tint. An inclined face in the light, as abhf, is lightest toward 
the observer, while an unillumined face tkdg is exactly the reverse. 

Notice that to give a flat effect on the inclined faces the spacing - 
out as also the change in the size of lines must be more gradual than 
when indicating curA-ature. (Compare with Figs. 46 and 50.) 



E-igr- S2, 




REMARKS ON SHAD IN G. — PLANE PROBLEMS. 27 

If two or more illuminated flat surfaces are i:)arallel to the paper (as tgb h, Fig. 52) but at 
different distances from the eye, the nearest is to be the lightest; if unilluminated, the reverse 
■would be the case. 

79. In treating of the theorj' of shadows distinctions have to be made, not necessary here, between 
real and apparent brilliant points and lines. We maj^ also remark at this point that to an experi- 
enced draughtsman some license is always accorded, and that he can not be expected to adhere 
rigidly to theor^y when it involves a sacrifice of effect. For example, in Fig. 46 we are unable to 
see to the left of the (theoretically) lightest jjart of the cylinder, and find it, therefore, advisable to 
move the darkest part past the point where, according to Fig. 43, we know it in reality to be. 
The professional draughtsmen who draw for the best scientific papers and to illustrate the circulars 
of the leading machine designers allow themselves the latitude mentioned, with most pleasing results. 
Yet until one may be justly called an expert he can depart but little from the narrow confines of 
theory without being in danger of producing decidedly peculiar effects. 

80. As from this point the student will make considerable use of the compasses, a few of the 
more important and frequentty recurring plane problems, nearly all of which involve their use, may well 
be introduced. The proofs of the geometrical constructions are in several cases omitted, but if desired 
the student can readily obtain them by reference to any s^'nthetic geometry or work on plane problems. 

All the problems given (except No. 20) have proved of value in shoji practice and architectural work. 

The student should again read Arts. 48-51 regarding special uses of the 30° and 45° triangles, 
which, with the T-rule, enable him to employ so many "draughtsman's" as distinguished from 
" geometrician's " methods ; also Arts. 36 and 37. 

81. Prob. 1. To draw a perpendicular to a given line at a given jmint, as A (Fig. 53), use the tri- 
angles, or triangle and rule as previously described ; or lav off equal distances Aa^ Ab, and with a and 
b as centres draw arcs ost, msn, with common radius greater than one -half ab. The required perpen- 
dicular is the line joining A with the intersection of these arcs. ., E"igr- ss- 




82. Prob. 2. To bisect a line, as M N, use its extremities 
exactly as a and b were employed in the preceding construc- 
tion, getting also a second pair of arcs (same radius for all 
the arcs) intersecting abwe the line at a point we may call x. 
The line from s to x wiU be a bisecting perpendicular. m- 

83. Prob. 3. To bisect an angle, as AVE, (Fig. 54), lay off on its sides any equal distances V a, 

x^ig-- s^. T' b. Use a and b as centres for intersecting arcs having a 

common radius. Join T^ with x, the intersection of these arcs, 
for the bisector required. 

84. Prob. If. To bisect an arc of a circle, as a m b (Fig. 54), 
bisect the chord acb by Prob. 2; or, bj^ Prob. 3, bisect the 
angle aVb which subtends the arc. 

85. Prob. 6. To construct an angle equal to a given angle, as 6 (Fig. 55), draw any arc a b with 
centre 0, then, with same radius, an indefinite arc m B, ^^s- ss. 
centre V; use the chord of a 6 as a radius, and from 
centre B cut the arc m B at x. Join V and x. Then 
angle AVB equals 6. 

86. Prob. 6. To pass a circle through three points, a, b 
and c, join them by lines ab, be, bisect these lines by '^'^ 
perpendiculars, and the intersection of the latter will be the centre of the desired circle. 





28 



THEORETICAL AND PRACTICAL GRAPHICS. 



87. Prob. 



I . 



To divide a 

s'igr- se. 






line into any nimiber of equal parts draw from one extremity as A 
(Fig. 56) a line A C making any random angle with the given line 
A B. "W^ith a scale point off on ^ C as many equal parts (size 
immaterial) as are required on AB; four, for example. Join the 
last point of division (4) with B; then parallels to such line from 
the other points will divide A B similarly. 

88. A secant to a curve is a line cutting it in two points. If 
the secant .4 -B be turned to the left about A the point B will approach A, and the line will pass- 
through A C and other secant positions. When B reaches and coincides ^^s- s'r. 
with A the line is said to be tangent to the curve. (See also Art. 368.) 

A tangent to a mathematical curve is determined by means of known 
properties of the curve. For a random or graphiccd curve the method illus- 
trated by Fig. 57 (a) is the most accurate and is as follows : Through T, the point of desired tan- 
gency, draw random secants to points on either side of it, as A, B, D, etc. 
and prolong them to meet a circle having centre T and any radius. On each 
secant lay off— from its intersection with the circle— the chord of that secant 
in the random curve. Thus, am=TA; hn=TB; pcl=TD. From s 
where the curve m n o p q cuts the circle, draw s T, which will be a tangent, 
since for it the chord has its minimum value. 

A normal to a curve is a line perpendicular to the tangent, at the point 
of tangency. In a circle it coincides in direction with the radius to the point of tangency. 

89. Prob. 8. To draw a tangent to a circle at a given point draw a radius to the point. The 
perpendicular to this radius at its extremity will be the required tangent. Solve with triangles. 

90. Prob. 9. To draio a tangent to a circle from a point without 
join the centre C (Fig. 58) with the given point A; describe a semi- 
circle on AC as a diameter and join A with D, the intersection of 
the arcs. ADC equals 90°, being inscribed in a semi -circle; AD 
is then the required tangent, being perpendicular to CD at its 
extremity. 

91. Prob. 10. To draio a tangent at a given point of a circidar ^ ^ iFig-- ss. 

arc whose centre is unknoivn or inaccessible, locate on the arc two points equidistant from the given 
point and on opjDOsite sides of it; the chord of these ijoints will be parallel to the tangent sought. 

92. A regular polygon has all its sides equal, as also its angles. If of three sides it is called 
the equilatercd triangle; four sides, the square; five, pentagon; six, hexagon; seven, heptagon; eight,, 
octagon; nine, nonagon or enneagon; ten, decagon; eleven, undecagon; twelve, dodecagon. 

The angles of the more imjaortant regular j)olygons are as follows : triangle, 
120°; square, 90°; pentagon, 72°; hexagon, 60°; octagon, 45°; decagon, 36°; 
dodecagon, 30°. The angle at the vertex of a regular polygon is the supplement 
of its central angle. 

93. For the polygons most frequently occurring there are many special 
methods of construction. All but the pentagon and decagon can be readily 
inscribed or circumscribed about a circle by the use of the T-rule and triangle. 
For example, draw a b (Fig. 59) with the T- rule, and c d perpendicular to it 
with a triangle. The 45 ° triangle will then give a square, acb d. The same triangle in two positions- 
would give ef and gh, whence ag, g c, etc., would be sides of a regular octagon. 





PLANE PROBLEMS. 



29 





94. The 60 ° triangle used as in Art. 51 would give -the hexagon ; and alternate vertices of the 
latter, joined, would give an equilgiefal triangle. Or the radius ?)f ,J;he circle stepped off six times on 
the circumference, and alternate'' iDoints connected, would result similarly. 

95. Prob. 11. An additional method for inscribing an equilateral triangle in a 
circle, ivhen one vertex of the triangle is given, as A, Pig. ^60, is to draw the diameter, 
A B, through A, and use the triangle to obtain the sides A C and A D, making 
angles of 30 ° with A B. D and C ^\•ill then be the extremities of the third side 
of the triangle sought. 

96. Prob. 12. To inscribe a circle in an .equilateral triangle 
draw a perpendicular fi-om any vertex to the oisposite side. The centre of the 
circle will be on such line, two -thirds of the distance fi-om vertex to base, while 
the radius desired will be the remaining third. (Fig. 61). 

97. Prob. IS. To insaibe a circle in any triangle bisect any two of the interior 
angles. , The intersection of these bisectors will be the centre, and its perpen- 
dicular distance from any side will be the radius ~of the circle sought. 

98. Prob. H. To inscribe a pentagon in a circle draw mutually perpendicular s'lg-- es. 
diameters (Fig. 62) ; bisect a radius as at s ; draw arc a x of radius s a and 
centre s; then chord ax=af, the side of the pentagon to be constructed. 

99. Prob. 15. To construct a regular polygon of any number of sides, the length 
of the side being given. 

Let A B (Fig. 63) be the length assigned to a side, and a regular polygon 
of x sides desired. Take x equal to nine for . illustration, draw a semi -circle with 
A B as radius and divide by trial into x (or 9) equal parts. Join B with x — 2 

points of division, or seven, beginning at A, and prolong all but the 
last. With 7 as a centre, radius A B, cut line B-6 at m by an 
arc, and join m with 7, giving another side of the required polygon. 
Using m in turn as a centre, same radius as before, cut B-5 (pro- 
duced) and so obtain a third vertex. 

This solution is based on the familiar principles (a) that if a 
regular polygon has x 
sides each interior angle 

equals ^^~ \ and (b) that the diagonals drawn 

from any vertex of the polygon make the same angles 
with each other as with the sides meeting at that vertex. 

100. Prob. 16. Another solution of Prob. 15. Erect a 
perijendicular HR (Fig. 64) at the middle point of the 
given side. With iJf as a centre, radius MS, describe 
arc SA and divide it by trial into six equal parts. Arcs 
through these points of di\'ision, using ^ as a centre, 
and numbered up from six, give the centres on the ver- 
tical line for circles passing through M and S and in 
which MS would be a chord as many times as the 
number of the centre. 

101. For any unusual number of sides the method 




S'igr- S3- 





30 



THEORETICAL AND PRACTICAL GRAPHICS. 




PLANE PROBLEMS. 



31 



by "trial and error" is often resorted to, and even for ordinary cases it is by no means to be 
despised. By it the di\'iders are set " by guess " to the probable chord of the desired arc, and, sup- 
posing a heptagon wanted, the chord is stepped off seven times around the circumference; care being 
taken to have the points of the di^dders come exactly on the arc, and also to avoid damaging the 
paper. If the seventh step goes past the starting point the di^dders require closing; if it falls short, 
the original estimate was evidently too small. Obviously the change in setting the di^dders ought 
in this case to be, as nearly as j^ossible, one -seventh of the error; and after a few trials one should 
" come out even " on the last step. 

102. Proh. 17. To lay off on a given circle an arc of the same length as a given straight line.^ Let 
t (Plate I, Fig. 1) be one extremity of the desired arc; ts the given straight line and tangent to 
the circle; tm equal one -fourth of ts, and sx drawn with centre m, radius m-s. Then the length 
of the arc tx is a close approximation to that of the line ts. 

103. Prob. IS. To lay off on a straight line the length of a given circular arc,^ or, technically, to 
rectify the arc, let af (Plate I, Fig. 3) be the given arc; ai the chord prolonged till fi equals one- 
half the chord af; and ae an arc drawn with radius ai, centre i. Then fe approximates closely 
to the length of the arc af. 

104. Proh. 19. To obtain a straight line equal in length to any given semi- circle,'' draw a diameter oh 
of the given semi -circle (Plate I, Fig. 2) and a radius inclined at an angle of 80° to the radius ch. 
Prolong the radius to meet the line b h k, drawn tangent to the circle at h. From k lay off the 
radius three times, reaching n. The line no equals the semi -circumference to four places of decimals. 

105. Prob. SO. To drcvw a circle tangent to two straight lines and a given circle. (Four solutions.) 
This problem is given more on account of the valuable exercise it will prove to the student in ab- 
solute precision of construction than for its probable practical appHcations. Fig. 4 (Plate I) illus- 
trates the geometrical principles involved, and in it a circle is required to contain the points s and 



1 These -metliods of approximation were devised by Prof, iiaukine. They are sufficiently accurate for arcs not exceeding 
60°. The error varies as the fourth power of the angle. The complete demonstr.ition of Prob. 17 can be found in the Philo- 
sophical Magazine for October, 1867, and of Prob. 18 in the November issue of the same year. 

2 In his Graphical Statics Cremona states this to be the simplest method known for rectifying a semi -circumference. Accord- 
ing to Bottcher it is due to a Polish Jesuit, Kochansky, and was published in the Acta Eruditorum Llpsiae, 16So. The demon- 
stration is as follows ; Calling the radius unity, the diameter would have the numerical value '2. 

Then in Fig. 2, Plate I, we have on = y/oh- -f lin- = \/ok- + {kn — k/i)- = \/4 -)- (3 — tan 30°)2 = 3.14159 -f 
The tangent of an angle (abbreviated to "tan.") is a trigonometric function whose numerical value can be obtained from 
a table. A draughtsman has such freguent occasion to use these functions that they are given here for reference, both as lines, 
and as ratios. 

Trigonometric Functions as Eatios. Trigonometric Functions as Lines. 



e = the given angle = CA B 

h = hypothenuse of triangle CAB 

a = A B = side of triangle adjacent to vertex of 9 

o = £C=side of triangle opposite to e 

Then sin . 9 = j- ; cos i — tt ; 



Cu-tangent of Q 



tan e = 



cotan Q - 



sm I 
' cos I 



: reciprocal of cosine. 



cos I 
' bin ( 



: reciprocal of tan 9. 




A'«^ B 

The prefix "co" suggests "complement;" the co-sine of B is the sine of the complement of 5, &c. As lines the functions- 
may be defined as follows : 

The sine of an arc (e. g., that subtended by angle 6 in the figure) is the perpendicular {C B) let fall from one extremity of 
the arc upon the diameter passing through the other extremity. If the radius A C, through one extremity of the arc, be 
prolonged to cut a line tangent at the other extremity, the intercepted portion of the tangent is called the tangent of the arc, 
and the distance, on such extended radius, from the centre of the circle to the tangent, is called the secant of the arc. 

The co-sine, co-secant and co-tangent of the arc are respectively the sine, secant and tangent of the complement of the- 
given arc. 



32 



THEORETICAL AND PRACTICAL GRAPHICS. 



a and be tangent to the line mv■^. Draw first any circle containing s and «, as the one called "aux. 
circle." Join « to a and prolong to meet mv^ at k. Prom k draw a tangent, k g, to the auxiliary- 
circle. With radius k g obtain m and i on the line m v. A circle through s, a and m, or through 
s, a and i will fulfill the conditions. For k g^ = k s X k a, as kg is a tangent and k s a, secant. 
But k i = k g, therefore k i^ ^=k s X k a, which makes k i a, tangent to a circle through s, a and i. 

In Fig. 5 (Plate I) the construction is closely analogous to the above, and the lettering identi- 
cal for the first half of the work. The "given circle" is so called in the figure; the given lines 
are P v and R v. Having drawn the bisector, v e, of the angle P v R, locate s as much below v e as 
a (the centre of the given circle) is above it, the line a s being perpendicular to v e. Draw v^m k i 
parallel to v p and at a distance from it equal to the radius of the given circle. Then s, a, k and 
VI V I of Fig. 5 are treated exactly as the analogous points of Fig. 4, and a circle obtained (centre d) 
containing a, s and i. The required circle will have the same centre d but radius d lo, shorter than 
the first by the distance w i. Treat s, a, and m, (Fig. 5), similarly, getting the smallest of the four 
possible circles. 

The remaining solutions are obtained by using the points a and s again, but in connection with 
a line y z parallel to v R and inside the angle, again at a ijerpendicular distance from one of the 
given lines equal to the radius of the given circle.* 

This problem makes a handsome plate if the given and required lines are drawn in black; the 
lines giving the first two solutions in red; the remaining construction lines m blue. 

106. Prob. 21. To draw a, tangent to two given circles (a problem that may occur in connecting 
^'igr- ev. band-wheels by belts) join their centres, c and o, (Fig. 67) 

and at s lay off s m and s n each equal to the radius of the 
smallei'- circle. Describe a semi-circle o h k c on o c as a 
diameter. Carry m and n to k and h, about o as a centre. 
Angles c k and c h o are each 90 °, being inscribed in a 
semi-circle ; and c ^ is parallel to a b, which last is one of 
the required tangents ; while c h is parallel to t x, a second 
tangent. Two more can be similarlj^ found. 
To unite two inclined straight lines by an arc tangent to both, radius given. Prolong 
the given lines to meet at a (Fig. 68). With a as a centre and 
the given radius describe the arc m n. Parallels to the given lines and 
tangent to arc m n meet at d, from which perpendiculars to the given 
lines give the points of tangency of the 
required arc, which is now drawn with 
the given radius. 
g n 108. Prob. 23. To drcm through a 

•given point a line lohich icill — if produced — pass through the inaccesssible 





f 



ll 




*Tliis solution is taken from Benjamin Alvord's Tangencies of Circles and of Spheres^ published by the Smithsonian Institute. 
Tliat valuable pamphlet presents geometrical solutions of tbe ten problems of Apollonius on tbe tangencies of circles, and also 
■of tbe fifteen problems on the tangencies of spheres, all of which are valuable to the draughtsman, both geometrically and as 
■exercises in precision. The solutions are based on the principle, illustrated by Fig. 57, that the tangent line or tangent curve 
is the limit of all secant lines or curves. The problems on the tangencies of circles are as follows, the number of solutions 
in each case being given : (1) To draw a circle through three points. One solution. (2) Circle through two points and tan- 
igent to a given straight line. Two solutions. (3) Circle through a given point and tangent to two straight lines. Two solu- 
tions. (4) Circle through two points and tangent to a given circle. Two solutions. (5) Circle through a given point, tangent 
to a given straight line and a given circle. Four solutions. (6) Circle through a given point and tangent to two given 
■circles. Four solutions. (7) Circle tangent to three straight lines, two only of which may be parallel. Four solutions. (8) Circle 
tangent to two straight lines and a given circle. Four solutions. (Art. 105, above). (9) Circle . tangent to two given circles and 
& given straight line. Eight solutions. (10) Circle tangent to three given circles. Eight solutions, 



PLANE PROBLEMS.— TAPERING CIRCULAR ARCS. 



33 




intersection of two lines. Join the given point e with any point ,/' on A B and also with some point 
g on CD. From anj' point h on A B draw h i parallel to / g, then i k parallel to g e and h k 
parallel to f e. The line k e will fulfill the conditions. 

~4. To drmo an oval upon a given line. Describe a circle on the gi\-en line, m n, (Fig 
70) as a diameter. 'With m and n as centres describe arcs, m .r, n x 
radius »i a. Draw m v and n t through v and t, the middle points 
of the quadrants y in, y n. Then m s and n r are the portions of 
m X and n x forming part of the oval. Bisect n c at q and draw q x. 
Also bisect c q at z and join the latter with x. Bisect y h in d and 
draw f d fi-om /', the intersection of n s and q x. Use / as a centre 
and f s as radius for an arc sk terminating on f d. The intersec- 
tion, h, of If with .1- : is then the next centre and /; k the radius 
of the are /.■ I which terminates on h y produced. The oval is then 

completed with ;/ as a centre and radius y I. The lower portion is symmetrical with the upper and 

therefore similarly constructed. 

110. Where exact tangencj' is the requirement novices occasionally endeavor to conceal a failure 
to secure the desired object by thickening the curve. Such a course usually defeats itself and makes 
more evident the error thev' thus hope to conceal. With such instruments of precision as the draughts- 
man employs there is but little, if anj', excuse for overlapping or falUng short. 

^^s- ''3.. i^ common error in drawing tangents where the lines are of appreciable 

thickness is to make the outsides of the lines touch ; whereas they should 
have their thickness in common at the point of tangency, as at T (Fig. 
70), where, evidentlj', the centre-lines a and b of the arcs would be exactly 
tangent, while the outer arc of M would come tangent to the inner arc of 
N. 
^Mlen either a tube or a solid cylindrical piece is seen in the direction of its axis the 
outline is, evidently, simply a circle; and often the only way to determine 
which of the two the circle rei^resented would be to notice which part of said 
end view was represented as casting a shadow. In Fig. 72, if the shaded 
arcs can cast shadows, the space inside the circles must be open, and the fig- 
ure would represent a portion of the end view of a boiler with its tubular 
openings. 

By exactly reversing the shading, the effect of which can be seen by turn- 
ing the figure upside down, it is converted into a drawing of a number of 
soUd, cylindrical jiins projecting from a plate. 

The tapering begins at the points where a diameter at 45° to the horizontal would cut the cir- 
cumference. 

To get a perfect taper on small circles use the bow-jjen and after making one complete circle 
add the extra thickness by a second turn which is to begin with the pen-point in the air, the jjen 
being brought down gradually upon the paper and then, while turning, raised fi-om it again. 

On medium and large circles the requisite taper can be obtained by a different process, viz., by 
using the same radius again but by taking a second centre, distant fi-om the first by an amount equal 
to the proposed width of the broadest part of the shaded arc • the line through the two centres to 
be perpendicular to that diameter which passes through the extremities of the taper. 




111. 










^^©i^ei^a 



^ 



34 



THEORETICAL AND PRACTICAL GRAPHICS. 



112. As an exercise in concentric circles Fig. 73 will prove a good test of skill. It is a fair 
representation of a gymnasium ring, the " annular torus " of mathematical works, and possessing 




some remarkable jiroperties, chief among which is the fact that it is the only surface of revokition 

known from which circles can be cut by three different systems of planes.* 

ng-. 7-4. E^igr- vs. 

Fitr X, / TORUS \ / 

C/ la y \D 




ip If 

113. In Fig. 74 the same surface is shown, in the centre, in outhne only. The axis of the 
surface would be a perpendicular to the paper at A. If MN represents a plane peirpendicular to 
the paper and containing the axis, then Fig. X will show the shape of the cut or section. As M N 
was but one of the positions of a plane containing the axis and as the surface might be generated 
by rotating MN with the circle ab about the axis, it is evident that one of the three systems of planes 
must contain the axis. 

When a surface can be generated by revolution about an axis one of its characteristics is that 
any plane perpendicular to the axis will cut it in a circle. The circles of Fig. 73 may then be, for the 
moment, considered as parallel cuts by a series of planes perjsendicular to the axis, a few of which 
may be shown in mn, op, &c. (Fig. X). Each of these planes cuts two circles from the surface; 
the plane o p, for example, giving circles of diameters c d and v iv respectively. 



* Olivier, Memoires de Geometrie Descriptive. Paris, 1851, 



. ANNULAR TORUS.— WARPED HYPERBOLOID. 35 

A plane perpendiciilar to the paper, on P Q, n'ould be a bi-iangent plane, because tangent to 
the surface at two points, P and Q; and such plane would cut two over-lapjDing circles from the 
torus, each of them running partly on the inner and partly on the outer portion of the surface. 
For the proof that such sections are circles the student is probaljly not prepared at this point, but 
is referred to OUvier's Seventh Memoir. 

114. Another interesting fact with regard to the torus is that a series of planes parallel to, but 
not containing the axis, cut it in a set of curves called the Cassian ovals, of which the Lenniiscate 
of Art. 158 is a special case, and which would result from using a plane parallel to the axis and 
tangent to the surface at a point on the smallest circle, as at a, (Fig. 74.)* 

115. Fig. Y is given to illustrate the fact that from mere untapered outlines, such as compose 
the central figure, we cannot determine the form of the object. By shading e/i/and DNr, the form 
shown in Fig. Y would be instantly recognized without the drawing of the latter. An angular object 
must therefore have shade lines, as also the end view of a round object ; but a side view of a cyl- 
indrical piece must either have uniform outlines or be shaded with several lines. 

Thus, in Fig. 76, A would represent an angular piece w^hile B would indicate a circular cylin- 
der ; if elliptical its section would be drawn at one side as shown. 



116. Before jjresenting the crucial test for the learner — the railroad rail— two additional practice 
exercises, mainly in ruling, are given in Figs. 77 and 78. The former shows that, like the parabola, 
the circle and hyperbola can be rejDresented by their envelojjing tangents. The upjDcr and lower 
figures are merely two views of the surface called the warped hyperboloid, from the hyperbolas which 
constitute the curved outlines seen in the upper figure. The student can make this surface in a 
few moments by stringing threads through equidistant holes arranged in a circle on two circular 
discs, of the same or different sizes, but having the same number of holes in each disc. By attaching 
weights to the threads to keep them in tension at all times, and giving the upper disc a twist, the 
surface will change from cylindrical or conical to the hyijerboloidal form shown. 

Gear wheels are occasionally constructed, having their teeth upon such a surface and in the 
direction of the lines or elements forming it; but the hyperboloid is of more interest mathematically 
than mechanically. 

Begin the drawing by iDcncilling the three concentric circles of the lower figure. When inking, 
omit the smaller circle. Draw a series of tangents to the inner circle, each one beginning on the mid- 
dle circle and terminating on the outer. Assume any vertical height, t s', for the u^jper figure, and 
draw H' M' and P' R as its upper and lower limits. H' M' is the vertical projection, or elevation, 
of the circle H K M N, and all points on the latter as 1,2,3,4, are projected, by perpendiculars to 
H' M', at 1', 2', 3', 4', etc. All points on the larger circle P QR are similarly projected to P' R'. The 
extremities of the same tangent are then joined in the ujjper view, as 1' with 1 (a). 

Part of each line is dotted to represent its disappearing upon an invisible portion of the sur- 
face. The law of such change on the lower figure is e\-ident from inspection ; while on the eleva- 



• These curves can also be obtained by assuming two foci, as if for an ellipse, but taking the product of the focal radii as 
a constant ciuantily, some perfect square. If pp' = 36" then a point on the curve would be found at the intersection of arcs 
having the foci as centres, and for radii 2" and 18", or 4" and 9", etc. When the constant assumed is the square of half the 
distance between the foci, the Lemniscate results. 



36 



THEORETICAL AND PRACTICAL GRAPHICS. 




a(«)2(n) Urt) 



32 31 




TAPERING LINES. — RAIL SECTIONS. 



37 



■FLs- 'na- 



tion the point of division on each line is exactly above the point where the other view of the same 
line runs through H M in the lower figure. 

117. To reproduce Fig. 78 draw first the circle afhn, then two circular arcs which would con- 
taiii a and b if extended, and whose greatest distance ft-om the original circle is x, (arbitrary). Six- 
teen equidistant radii as at a, t, d, etc., are next m order, of which the rule and 45° triangle give 
those through a, d,f and h. At their extremities, as m and n. lay off the desired width, y, and draw 
toward the points thus determined lines radiating from the centre. Terminate these last upon the inner 
arcs. Ink by drawing /Vo?7i the centre, not through or toward it. 

All construction lines should be erased before the tapering lines are filled in. The " filling in " 
may be done very rapidly by ruling the edges of the line in fine at first, then opening the pen 
slightly and beginning again whei-e the opening between the lines is apparent and ruling fi-om there, 
adding thickness to each edge on its inner side. It will then be but a moment's work to fill in, free- 
hand, with the Falcon jjen or a fine-pointed sable-brush, between the now heavy edge-lines of the 

tapier. To have the pen make a coarse line 
when starting from the centre would destroy 
the effect desired. 

118. The draughtsman's ability can scarcely 
be put to a severer test on mere outline work 
than in the drawing of a railroad rail, so many 
are the changes of radii involved. 

As previously stated, where tangencies to 
straight lines are required, the arcs are to be 
f drawn first, then the tangents. 

Figs. 79 and 80 are photo-engravings of 
rail sections, showing two kinds of "finish." 
Fig. 80 is a "' working drawing " of a Penn- 
sylvania Railroad rail, full-size. If finished with 
shade lines, as in Fig. 79, section-lined with 
Prussian blue, and the dimension Unes drawn 
in carmine this makes one of the handsomest 
plates that can be undertaken. 

A still higher effect is shown in the wood- 
cut of the title-page, the rail being represented in oblique projection and shaded. 

Begin Fig. 80 by drawing the vertical centre-hne, it being an axis of symmetry. Upon it lay 
off 5" for the total height, and locate two points between the top and base at distances from them 
of II" and I" respectively; these to be the points of convergence of the lower lines of the head and 
sloping sides of the base. From these points draw lines, at first indefinite in length, and inclined 
13° to the horizontal. The top of the head is an arc of 10" radius, subtended by an angle of 9°. 
This changes into an arc of -f^" radius on the upper corner, with its centre on the side of said 9° 
angle. The sides of the head are straight lines, drawn at 4° to the vertical and tangent to the corner arcs. 
The thin vertical portion of the rail is called the iveh and is y^" wide at its centre. The outlines of the 
web are arcs of 8" radius, subtended by angles of 15°, centres on line marked "centre line of bolt holes." 
The weight per yard of the rail shown is given as eighty-five pounds, from which we know the 
area of the cross-section to Ije eight and one-half square inches, since a bar of iron a yard long 
and one square inch in cross section weighs, approximately, ten pounds. (10.2 lbs., average.) 




38 



THEORETICAL AND PRACTICAL GRAPHICS. 



The proportions given are slightly different from those recommended in the report* of the com- 
mittee appointed by the American Society of Civil Engineers to examine into the proper relations to 




each other of the sections of railway wheels and rails. There was quite general agreement as to 

the following recommendations: a top radius of twelve inches; a quarter-inch corner radius; vertical sides 

s'igr- eo- I 

2^ IN. 




to the web ; a lower-corner radius of one-sixteenth inch, and, lastly, a broad head relatively to the depth. 



•Transactions A. S. C. E. January, 1891. 



EXERCISES FOR THE IRREGULAR CURVE. 



39 



CHAPTER V. 



THE HELIX.— CONIC SECTIONS.— HOMOLOGICAL PLANE CUEYES AND SPACE-FIGURES.— LINK-MOTION 
CURVES.-CENTROIDS.— THE CYCLOID.— COMPANION TO THE CYCLOID.- THE CURTATE TRO- 
CHOID.-THE PROLATE TROCHOID.— HYPO-, EPI-, AND PERI-TROCHOIDS.— SPECIAL TROCHOIDS— 
ELLIPSE, STRAIGHT LINE, LIMA^'ON, CARDIOID, TRISECTRIX, INVOLUTE, SPIRAL OF ARCHI- 
MEDES. -PARALLEL CURVES— CONCHOID.— QUADEATRIX.— CISSOID.— TEA CTRIX.— WITCH OP 
AGNISI.-CARTESIAN OVALS.— CASSIAN OVALS.- CATENARY.— LOGARITHMIC SPIRAL.— HYPER- 
BOLIC SPIRAL.-THE LITUUS. 



119. There are many curves which the draughtsman has frequent occasion to make whose con- 
struction involves the use of the irregular curve. The more important of these are the Helix; Conic 
Sections — Ellipse, Parabola and Hjq^erbola ; Link-motion curves or point-paths ; Centroids ; Trochoids ; the 
Involute and the Spiral of Archimedes. Of less practical importance, though ecjually interesting 
geometrically, are the other curves mentioned in the heading. 

The student should become thoroughlj^ acquainted with the more important geometrical proper- 
ties of these curves, both to facilitate their construction under the varying conditions that may arise 
and also as a matter of education. Considerable sjDace is therefore allotted to them here. 

At this point Art. 58 should be reviewed, and in addition to its suggestions the student is fur- 
ther advised to work, at first, on as large a scale as possible, not undertaking small curves of sharp 
curvature until after acquiring some facility in the use of the curved ruler. 

THE HELIX. 

120. The ordinarj' helix is a curve which cuts all the elements of a cyhnder at the same angle. 
Or we maj' define it as the curve which would be generated by a point having a uniform motion 
around a straight hue combined with a uniform motion parallel to the line. 



X'ig-. SI. 




ffluuuuuuuuumuumuui 

The student can readily make a model of the cylinder and helix by 
drawing on thick paper or Bristol-board a rectangle A" B" 0" D" (Fig. 81) 
and its diagonal, D" B" ; also equidistant elements, as m" 6," n" c" etc. Allow 
at the right and bottom about a quarter of an inch extra for overlapping, as 

shown by the lines x y and .s z. Cut out the rectangle z x ; also cut a series of vertical 
slits between D" C" and zs and turn the divisions up at right angles to the paper; put mucilage 
between B" C" and x ij ; then roll the paper up into cylindrical form, bringing A" D" t" ft" in front 



40 THEORETICAL AND PRACTICAL GRAPHICS. 

of and upon the gummed i^ortion, so that A" D" will comcide with B" C". The diagonal D" B" will 
then be a helix on the outside of the cylinder, but half of which can be seen in front view, as D 
7', (see right-hand figure) ; the other half, 7' A', being indicated as unseen. 

To give the cylinder more permanent form it can be pasted to a cardboard base by mucilage 
on the under side of the divisions turned ujd along its lower edge. 

The rectangle A" B" C" D" is called the development of the cylinder ; and any surface like a cyl- 
inder or cone, which' can be rolled out on a plane surface and its equivalent . area obtained by 
bringing consecutive elements into the same plane, is called a developable surface. The elements wi" 6", 
n" c", etc., of the development stand vertically &i b, c, d . . . . g of the half plan, and are seen in the 
elevation at m' b\ n' d, d d', etc. The point 3', where any element, as c', cuts the helix, is evidently as 
high as o" where the same point appears on the development. We may therefore get the curve ff 7' A' 
by erecting verticals from b, c, d, . . . g, to meet horizontals from the points where the diagonal D" 
B" crosses those elements on the development. 

The length D" C" obviously equals 2 tt r, in which r = D. 

The height A' ly, that the curve attains in winding once around the cylinder, is called the pitch 
of the helix. 

The construction of the helix is involved in the designing of screws and screw-propellers, and in 
the building of winding stairs and skew-arches. 

Mathematically the helix and its orthographic ijrojection are well worth' study, particularly the 
latter when the helix crosses the elements at 45° ; it becoming then identical with Roberval's curve 
of sines, otherwise known as the companion to the cycloid. 

For a helix on a cone refer to the article on the Spiral of Archimedes. 

THE CONIC SECTIONS. 

121. The ellipse, parabola and hyjDerbola are called conic sections or conies because they may be 
obtained by cutting a cone by a plane. We will, however, first obtain them by other methods. 

According to the definition given by Boscovich, the ellipse, parabola and hyperbola are curves in 
which there is a constant ratio between the distances of laoints on the curve fi:om a certain fixed 
point (the focus) and their distances from a fixed straight line (the directrix). 

Referring to the parabola, Fig. 82, if S and B are points of the curve, F the focus and A' Y the 
directrix, then, if /S F: S T: : B F: B X, we conclude that B and S are points of a conic section. 

122. The actual value of the ratio (or eccentricity) may be 1 or either greater or less than unity. 
When SF equals ST the ratio equals 1, and the relation is that of equality, or parity, which sug- 
gests the parabola. 

123. If it is farther from a point of the conic to the focus than to the directrix the ratio is 
greater than 1 and the hyperbola is indicated. 

124. The ellipse, of course, comes in for the third possibility as to ratio, viz., less than 1. Its 
construction by this principle is not shown in Fig. 82 but later, the method of generation here 
given illustrating the practical way in which, in landscape gardening, an elliptical plat would be laid 
out; it is therefore called the construction as the "gardener's ellipse." 

Taking A C and D E a,s representing the extreme length and width, the points F and Fj^ ifoci) 
would be found by cutting ^ C by an arc of radius equal to one-half A C, centre D. Pegs or pins 
at F and Pj, and a string, of length A B, with ends fastened at the foci, complete the preUminaries. 
The curve is then traced on the ground by sliding a pointed stake against the string, as at P, so 
that at all times the parts -F, P, F P, are kept straight. 



CONIC SECTIONS. 



41 



125. According to the foregoing construction the elUpse may be defined as a cvirve in which the 
sum of the distances from any point of the curve to tivo fixed points is constant. That constant is evidently 
the longer or transverse (major) axis, A C. The shorter or conjugate (or minor) axis, D E, is perperL- 
dicular to the other. 

With the comijasses we can determine P and other points of the ellipse by using F and F^ as 
centres, and for radii any two segments of A C. Q, for example, gives ^1 Q and C Q as segments. 
Then arcs from F and F^, with radius equal to Q C, would intersect arcs from the same centres, 
radius Q A, in four points of the ellipse, one of which is P. 

126. By the Boscovich definition, we are enabled to construct the parabola and hyperbola also 
by continuous motion along a string. 

For the parabola place a triangle as in Fig. 82, with its altitude G X toward the focus. If a 
string of length G X be fastened at G, stretched tight from G to any point B, by putting a pencil 
at B, then the remainder B X swung around and the end fastened at F, it is then, evidently, as 
far from B to F as it is from B to the directrix; and that relation will remain constant as the tri- 
angle is slid along the directrix, if the pencil point remains against the edge of the triangle so that 
the portion of the string fi-om G to the pencil is kept straight. 



^ig-- ©2. 




127. For the hyperbola, (Fig. 82), the construction is identical with the i^receding, except that 
the string fastened at / runs down the hypothenuse and equals it in length. 

128. Referring back to Fig. 35, it will be noticed that the focus and directrix of the jjarabola 
are there omitted ; but the former would be the point of intersection of a perpendicular from A 
upon the line joining C with E. A line through A, parallel to O E, would be the directrix. 

129. Like the ellipse, the hyperljola can be constructed by using two foci, but whereas in the 
ellipse (Fig. 82) it was the sum of two focal radii that was constant, i. e., FP+FiP=FD + 



42 



THEORETICAL AND PRACTICAL GRAPHICS. 




F^D = A C (the transverse axis), it is the difference of the radii ^'^s- sa. 

that is constant for the hyperbola. 

In Fig. 83, if J. -B is to be the transverse axis of the 
two arcs, or "branches," which make the comijlete hyperbola, 
then using p and p to represent any two focal radii, as F Q 
and Fi Q, or FR and F^ R, we will have p — p = A B (the 
constant quantitjO- 

To get a point of the curve in accordance with this jsrin- 
cii^le we may lay off from either focus, as F, any distance ■ 
greater than F B, as F J, and with it as a radius, and F as 
a centre, describe the indefinite arc JR. Subtracting the con- 
stant, A B. from F J, by making J E = A B, we use the 
remainder, F E, as a radius, and F-^ as a centre to cut the 
first arc at R. The same radii will evidently determine three 
other jioints fulfilling the conditions. 

130. The tangent to the hyperbola at any point, as Q, bisects the angle F Q F^, between the 
focal radii. 

In the ellipse, (Fig. 82), it is the external angle between the radii that is bisected by the tan- 
gent. 

In the jaarabola, (Fig. 82), the same principle applies, but as one focus is supposed to be at 
infinity, the focal radius, B G, toward the latter, from any point, as B, would be parallel to the axis. 
The tangent at B would therefore bisect the angle F B X. 

131. The ellipse as a circle vieiuecl obliquely. If ARMBF (Fig. 84) were a circular disc and we 

were to rotate it on the diameter A B, it would become 
narrower in the direction FE until, if sufficiently turned, 
only an edge view of the disc would be obtained. The 
axis of rotation, A B, would, however, still appear of its 
original length. In the rotation supposed, all points 
not on the axis would describe circles about it with 
their planes jDerpendicular to it. M, for example, 

1 would move in the arc of a circle part of which is 
shown in M M-^ , which is straight, as the plane of the 
arc is seen " edge-wise." If instead of a circular disc 
we were to take one of elliptical form, as A C B D, and 
turn it upon its shorter axis CD, it is obvious that B 
would apparently approach on one side while A ad- 
vanced on the other; and when B was directly in front 
of, and projected at k, we would have the ellipse projected in the small circle r t k. Having given 
then the two axes of a desired ellipse, as ^ -B and C D, use them as diameters of concentric circles 
and from their common centre, 0, draw random radii, as M, T, OK. Where any radius, M, 
meets the outer circle, drojD a perjaendicular M M^ to the longer axis, to meet a line m M^ , parallel 
to the same longer axis and passing through m, where M cuts the smaller circle. 

132. If rS is a tangent at T to the large circle, then when T appears at T^ we shall have T^ 
S as a, tangent to the ellipse at the point derived from T ; S having remained constant, being on 
the axis of rotation. 




CONIC SECTIONS. 



43 




Similarly, if a tangent at R^ were wanted we may first find r, corresponding to R^; draw the 
tangent- ?•/ to the small circle; then join J, (constant point), with R^. 

133. Occasionally we have given the length and inclination of a pair of diameters of the elhpse, 
^ig-. as. making oblique angles with each other. Such diameters 

N 

are called conjugate and the curve may be constructed 
iTpon them thus: Draw the axes TD and H K ai the 
assigned angle D H; construct the parallelogram M N 
X Y ; divide D M and D into the same number of 
equal parts ; then irom K draw lines through the points 
of division on D 0, to meet similar lines through H and 
the divisions on D M. The intersection of like nimibered 
lines will give points of the elliiDse. 

134. It is the law of expansion of a j^erfect gas that the volume is inversely as the pressure. 
That is, if the volume be doubled the pressure drops one-half; if trebled the pressure becomes one- 
third, etc. Steam not being a perfect gas departs somewhat from the above law, but the curve in- 
dicating the fall in pressure due to its expansion is compared with that for a perfect gas. 

To construct the curve for the latter let us suppose C L K G (Fig. 86) to be a cylinder with a 
volume of gas C G b c behind the piston. Let c h indicate the ^^er- ss. 

pressure before expansion begins. If the piston be forced ahead 
by the expanding gas until the volume is doubled, the pressure 
will drop, by Boyle's law, to one-half and will be indicated by 
t d. For three volumes the j)ressure becomes vf, etc. The curve 
CSX is an hyperbola, the special case called equilateral. 

Suppose the cylinder were infinite in length. Since we cannot 
conceive a volume so great that it could not be doubled, or a pressure so smaU that it could not 
be halved, it is evident that theoretically the curve c x and the line G K will forever ajjproach each 
other yet never meet ; that is, they will be tangent at infinity. In such a case the straight line is 
called an asymptote to the curve. 

135. Although the right cone (i. e., one having its axis perpendicular to the plane of its base) 
is usually employed in obtaining the ellipse, hyperbola and parabola, yet the same kind of 

-tr sections can be cut from an oblique or scalene cone of circular l^ase, 
as T'! A B, Fig. 87. Two sets of circular sections can also be cut 
from such a cone, one set, obviously, by planes parallel to the base, 
the other by planes like CD, making the same angle with the lowest 
element, V B, that the highest element, V A, makes with the base. 
The latter sections are called sub-contrary. 

To prove that the section x y is a, circle we note that both it 
and the section m n — the latter known to be a circle because parallel 
to the base — intersect in a line perpendicular to the paper at o. 
This iine pierces the front surface of the cone at a point we may 
call r. It would be seen as the ordinate o r (Fig. 88), were the fi-ont half of circle 
m n rotated until parallel to the paper. Then o r'' = o m X o n. But fi-om the '^^ 
similar triangles in Fig. 87 we have o m : o y :: o x : o n whence oy X o x = o mX on 
= or', thus showing that the section xy ia circular. 

Were the vertex of a scalene cone removed to infinity the cone would become an oblique cylinder 
with circular base; but the latter would possess the property just established for the former. 





iF-igr- es. 
o 




44 



THEORETICAL AND PRACTICAL GRAPHICS. 



136. The most interesting practical application of the sub-contrary section is in Stereographic 
Projection, one of the methods of representing the earth's surface on a ^^s-- ss. 
map. The especial convenience of this projection is due to the fact 
that in it every circle is projected as a circle. This results from the 
relative position of the eye (or centre of projection) and the plane of j^ 
projection; the latter is that of some great circle of the earth and the 
eye is located at the pole of such circle. 

137. In Fig. 89 let the circle ABE represent the equator; MN 
the plane of a meridian, also taken as the plane of projection; A B j. 





any circle of the sphere; E the position of the eye: then a b, the projection of ^ 5 on plane M N^ 



CONIC SECTIONS. 



45 



(see Art. 4), is a circle, being a sub-contrary section of the visual cone E. A B, as the student 
can easilj' prove. 

138. We now take up the conic sections as derived from a right cone. 

A complete cone (Fig. 90) lies as much above as below the vertex. To use the term adopted 
from the French, it has two nappes. 

Aside from the extreme cases of perpendicularity to or containing the axis the inclination of a plane 
cutting the cone may be 

(a) Equal to that of the elements,* therefore jiarallel to one element, giving the parabola, as M 
Q l/j (Fig. 90) ; the plane k q M being parallel to the element V U and therefore maktag with the 
base the same angle, 0, as the latter. 

(b) Greater than that of the elements, causing the plane to cut both nappes and giving a two- 
branch curve, the hyperbola, as DJE and fhg (Fig. 90). 

(c) Less than that of the elements, the plane therefore cutting all the elements on one side of the 
vertex, giving a closed curve, the ellipse; as KsH, Fig. 91. ^^s- si- 

139. Figures 90 and 91, with No. 4 of Plate 2 are 
not only stimulating examples for the draughtsman Ijut 
they illustrate probably the most interesting fact met 
with in the geometrical treatment of conic sections, viz., 
that the spheres which are tangent simultaneously to the 
cone and the cutting planes, touch the latter in the foci 
of the conies ; while in each case the directrix of the 
curve is the line of intersection of the cutting plane 
and the plane of the circle of tangency of cone and 
spihere. 

To establish this we need only emjjloy the well known 
princijjles that (a) all tangents fr'om a jjoint to a 
sphere are equal iu length, and (b) all tangents are 
equal that are drawn to a sphere fr-om jjoints equidis- 
tant from its centre. In both figures all jjoints of the 
cone's bases are evidently ec[uidistant fr-om the centres 
of the tangent spheres. 

140. On the ujai^er naj^ije, (Fig. 90), let S H he the circle of contact of a sphere which is tan- 
gent at F^ to the cuttmg i)lane P L K. The plane P H^, of the cfrcle cuts the plane of section in 
P m. If D is any point of the curve D J E, J another point, and we can jjrove the ratio constant 
(and greater than unity) between the distances of D and J fr-om F^ and their distances to P m, 
then the curve DJE must be an hyperbola, by the Boscovich definition ; F^ must be the focus and 
P m the directrix. 

D F^ is a tangent whose real length is seen at X S. J F^ and / S are equal, being tangents to 
the sphere from the same point. We have then the proportion XS:GR::.IS:JR or D F^: D Z:: 
J F^: J R. Since / S and its equal .7 F^ are greater than / R, and the ratio J F^ to J R is constant, the 
proposition is established. 

141. For the jiarabola on the lower napj^e, since the plane Mek is inclined at the angle & 
made by the elements, we have Q A= Q B (opposite equal angles) and Q B equal Q F (equal tan- 
gents). M F=B W=Mo, therefore M F: M o:: Q F: Q o, and it is as far fr-om M to the focus F as 
to the directrix s x, fulfilling the condition essential for the parabola. 




•See Eemark, Art. 4. 



46 



THEORETICAL AND PRACTICAL GRAPHICS. 



142. For the ellipse K s H, (Fig. 91), we have PX and iVT as the lines to be i^roven direc- 
trices, and F and F^ the points of tangency of two spheres. Let s be any point of the curve under 
consideration and VL the element containiiag s. This element cuts the contact circles of the spheres 
in a and A. A plane through the cone's axis and jiarallel to the paper would contain o t, o v and 
V n. Prolong v oi to meet a line V B that is parallel to K H. Join E with a, producing it to meet 
PX at r. In the triangles asr and aVR we have sa:sr:: V a: V R. But sa = sF (equal tangents) 
and similarly Va= Vn; therefore sF:87-:: Vn: V R, which ratio is less than unity; therefore a is a 
point on an ellipse.* 

The ijlane of the intersecting lines V a and R r cuts the plane M iV in AT which is therefore 
parallel to ar; therefore sA:s T:: Vn: V R. But s A = s F^ therefore s F^: s Tr. Vn: V R, the same 
ratio as before. 

143. If the plane of section P N were to approach parallelism to F C the point R would ad- 
vance toward n, and when VR became Vn the plane would have reached the position to give the 
parabola. 



S2. 




* Schlomilcli, Geometrie des Maasses, 1874. 



CONIC SECTIONS AS HOMOLOGOUS FIGURES. 



47 



144. The proof that KsH (Fig. 91) is an elliiDse when the curve is referred to two foci is as 
follows : KF= K m ; KF, = Kt; therefore K F + K F, = K m + Kt= t m = x n = S K F + F F, = 
'2HF+FF,; i. e., K F = H F,. 

Since s F= s a and s F^ — s A we have s F + s F^ = s a + s A = A a = im = x n = H K. The 
sum of the distances from any point s to the two fixed points F and F^ is therefore constant and 
equal to the longer axis H K. 

HOMOLOGOUS PLANE AND SPACE FIGURES. 

145. Before leaving the conic sections their construction will be given by the methods of Pro- 
jective Geometry. (See Art. 9.) 

In Fig. 92, if S^ is a centre of projection, then, by Art. 4, the figure A^ B^ C" is the central 
projection of A^ B^ Cy The points A^ and A^ are corresponding points. Similarly B^ and B^, C^ and C^. 

For ^3 as the centre of projection the figure A^ B^ C[ corresponds to A^ B^ C^. 

If we join S, with S^ and prolong to 0, to meet the plane of the figures A^ B^ C, and A.^ B^ C„ 
we find that the jjoint sustains the same relation to these two figures that S^ and S„ do to the 
figures just taken in connection with theni. Using the technical term trace for the intersection of a 
line with a jjlane or of one plane with another, we see that A^ c^ is the trace, on the vertical j^lane, 
of any plane containing A^ B^. This plane cuts the " axis of homology," t^ m, in c^. As A^ B^ lies 
in the jjlane of S^ and A^ 5', and in the horizontal j^lane as well, it can only meet the vertical plane 
in c„, the point of intersection of all these jjlanes. Similarly we find that A^ C" and A^ C^, if 
prolonged, meet the axis at the same point; correspondingly B'^ C- and B^ C^ meet at a„. But A^ B'^ 
and A^ B^, being corresponding lines, lie in the plane with S^, though belonging to figures in two 
other planes; they must, therefore, meet also at the same point, c„; and similarly for the other 
lines in the figures used with S.^- ^i-s- ©3. 

Finally, the line A^ A^ being the hor- 
izontal trace of the plane determined by 
the hnes joining A^ with S^ and S^ must 
contain the horizontal trace, 0, of the line 
joining S^ with S^ But this puts A,, and 
A^ into the same relation with that A^ 
and A' sustain to S^; or that of A^ and 
A^ to »S'i. Hence we rightly conclude that 
on one plane we can take a point as a 
centre of projection, and a figure A^ B, G,, 
and from them derive a second figure, ^Ij 
JBi C, , which corresponds to the assumed 
figure in the same way as if they lay in 
different planes. Figures so related in one 
plane are called homological figures and the 
centre, 0, a centj-e of homology. 

146. Had Ai B^ C, been a circle, and all 
its points joined with S^, then from what has 
preceded we know that A^ 5' C would have 
been an ellipse; as also would have been 
the case were A.^ B^ C„ a circle used in con- 
nection with (Sj. But our conclusions should enable us to substitute a circle for A^ B^ C„ and using 




48 



THEORETICAL AND PRACTICAL GRAPHICS. 



in the same plane with it get an ellipse in place of the triangle A^Bi Cj. Before illustrating this 
it is necessary to show the relation of the axis to the other elements of the problem and to supply 
a test as to the nature of the conic. 

147. First as to the axis, and employing again for a time a space figure (Fig. 93), it is evident 
that raising or lowering the horizontal plane c X Y jDarallel to itself, and with it, necessarily, the 
axis, would not alter the kind of curve that it would cut from the cone S. H A B, were the elements 
of the latter prolonged. But raising or lowering the centre, S, would decidedly affect the curve. 
Where it is, there are two elements of the cone, S A and S B, which would never meet the plane 
c X Y. The shaded plane containing those elements meets the vertical plane in " vanishing line (a)," 
parallel to the axis. This contains the projections, A and B, of the f)oints at infinity where the 
lower i^lane may be considered as cutting the elements SA and SB. Were S and the shaded f)lane 
raised to the level of H, so that " vanishing line (a) " should become tangent to the base, there would 
be one element, S H, of the cone, parallel to the lower plane, and the section of the cone by the latter 
would be the parabola; while the figure as it stands would indicate the hyperbola. The former 
would have but one jjoint at infinity; the latter, two. 

148. Raise the centre S so that the vanishing line does not cut the base and, evidently, no Hne 
from S to the base would be parallel to the lower plane; but the latter would cut all the elements 
on one side of the vertex, giving the ellipse. 

149. Bearing in mind that the projection of the circle A H B t is on the lower plane produced, 
if we wish to bring both these figures and the centre S into one plane, without destroying the relation 
between them, we may imagine the end plane Q L X removed and rotation of the remaining system 




occurring about cr ^ in a manner exactly similar to that which would occur were i oj c a system of 
four pivoted links and the point o pressed toward c. The motion of S would be parallel and equal 



CONIC SECTIONS AS HOMOLOGOUS FIGURES. 



4& 



to that of 0, and, like the latter, would evidently maintain its distance from the vanishing line and 
describe a circular arc about it. The vanishing line would remain parallel to the axis. 

150. From the foregoing we see that to oljtain the hyperbola by projection of a circle from a 
point in the plane of the latter we would require simply a secant vanishing line, MX (Fig. 94),, 
and an axis of homology parallel to it. Take any point P on the vanishing line and join it with 
any point K of the circle. P K meets the axis at y ; therefore the line ky that corresponds to P K 
must also meet the axis at i/. P is analogous to S ^ of Fig. 93 in that it meets its corresponding 
line at infinity, i. e., is parallel to it. Therefore y k, parallel to P meets the ray K at k. cor- 
responding to K. Then A' joined with an}' other point R gi\-es K z. .Join z with the point k 
just obtained aiid jKolong R io intersect them, obtaining r, another point of the hyperbola. 

151. In Fig. 93, were a tangent at B drawn to arc A H B, it would meet the axis in a jjoint 
which, like all points on the axis, corresjjonds to itself. From that point the projection of that 
tangent on the lower i^lane would be parallel to S B, since they are to meet at infinit\'. Or if S J 
is parallel to the tangent at B then ./ will be the projection of J' at infinity, where SJ meets 
the tangent; / will be therefore one jDoint of the projection of said tangent on the lower plane;, 
while another point would be, as previousl}' stated, that in which the tangent at B meets the axis. 

152. Analogously in Fig. 94, the tangents at M and N meet the axis, as at F and E; but the 
projectors OM and ON go to points of tangency at infinity; M and N are on a "vanishing line"; 
hence M is parallel to the tangent at infinity, that is to the nsymptoie (see Art. 134) through F; 
while the other asymptote is a parallel through E to X. 

153. As in Fig. 93 the projectors from .S' to all points of the arc above the level of S could 
cut the lower plane only by being produced to the right, giving the right-hand branch ol the hyper- 
bola; so, in Fig. 94, the arc MUX, above the vanishing line, gives the lower branch of the hyperbola. 
To get a point of the latter, as h, and having already obtained any point x of the other Ijranch,- 
join H with X (the original of x) and get its intersection, r/, with the axis. Then x g h corresponds 
to g X H, and the ray OH meets it at h, the pirojection of H. 

The cases should be worked out in which the vanishing line is tangent to the circle or exterior to it. 

154. The homological figures with which we have been dealing were plane figures. But it is- 
possible to have space figures homological with each other. 

In homological space figures correspioncling lines meet 
at the same point in a plane, instead of the same point 
on a line. A vanishing pAanc takes the jilace of a \-an- 
ishing line. The figure that is in homology with tlie 
original figure is called the relief-per-fpective of the latter. 
(See Art. 11.) 

Remarkafily beautiful effects can be obtained by the 
construction of homological space figures, as a glance at 
Fig. 95 will show. The figure represents a triple row 
of groined arches and is ii-om a photograph of a model 
designed Ijy Prof. L. Burmester. 

Although not always requiring the use of the irregular curve and tlierefore not strictly the material 
for a to})ic in this chapter, its close analogy to the foregoing matter may justify a lew words at this 
point on the construction of a relief-jiersijective. 

155. In Fig. 96 the plane P Q is called the plane of homology or pidure-plane, and — adopting 
Cremona's notation — we will denote it by tt. The vanishing plane, M N, or ^', is parallel to it. is 



■Fi-s- ©5- 




50 



THEORETICAL AND PRACTICAL GRAPHICS. 



the centre of homology or perspective-centre. All points in the plane n are their own perspectives or, in 
other words, corresimid to themselves. Therefore B" is one point of the projection or perspective of 
the line A B, being the intersection of A B with tt. The line v, parallel to A B, would meet the 




latter at infinity; therefore v, in the vanishing plane, 4>', would be the projection upon it of the 
point at infinity. Joining v with B" and cutting v B" by rays OA and OB gives A' B' as the relief- 
perspective of A B. The plane through and A B cuts tt in B" n, which is an axis of homology for 
AB and A' B', exactly as mm in Fig. 92 is for A, B, and A, B,. 

Fig-- S'7- 






A 





C->i° 



\-- 



As D C is parallel to A B (Fig. 96) a parallel to it through is again the line v. 




LINK-MOTION CURVES. 51 

The trace of D C on -n- is C". Joining v with ('" and cutting v C" by rays D, C, obtains 
IXC in the same manner as A' B' was derived. The originals of A' B' and C Z)' are parallel lines; 
but we see that their reliet-perspectives meet at v. The vanishing plane is therefore the locus* of 
the vanishing points of lines that are parallel on the original object; while the plane of homology 
is the locus of the axes of homology of corresponding lines ; or, differently stated, any line and its 
relief-persiJective will, if produced, meet on the plane of homolog}'. 

156. Fig. 97 is inserted here for the sake of completeness, although its study may be reserved, if 
necessary, until the chapter on projections has been read. In it a solid object is represented at 
the left in the usual views, plan and elevation ; G L being the ground line or axis of intersection 
of tlie planes on which the views are made. The planes tt and <^' are interchanged, as compared 
with their positions in Fig. 96, and they are seen as lines, lieing assumed as perpendicular to the 
paper. The relief-perspective appears between them, in plan and elevation. 

The lettering of A B and D C, and the lines employed in getting their relief-perspectives, being- 
identical with the same constructions in Fig. 96 ought to make the matter clear at a glance to all 
who have mastered what has preceded. 

Burmester's Grundzi'igc der Relief- Perspective and \\'iener's Darstellende Geomctrie are valuable reference 
works on this topic for those wishing to pursue its studj^ further; ).)ut for special work in the 
line of homological plane figures the student is recommended to read Cremona's Projective Geometry 
and Graham's Geometry of Position, the latter of which is especially \'aluable to the engineer or architect 
since it illustrates more fully the practical application of central projection to Graphical Statics. 

LINK-MOTIOX CURVES. 

167. Kineiiiatics is the science which treats of j^ure motion, regardless of the cause or the results of 
the motion. 

It is a i^urely kinematic problem if we laj' out on the drawing-board the path of a point on 
the connecting-rod of a locomotive, or of a point on the jDiston of an oscillating cylinder, or of any 
point on one of the moving pieces of a mechanism. Such problems often arise in machine design, 
especiallj' in the invention or modification of valve-motions. 

Some of the motion-curves or point-paths that are discovered by a study of relative motion are 
without siaecial name. Others, whose mathematical properties had already been investigated and the 
curves dignified with names, it was later found could be mechanically traced. Among these the 
most familiar examples are the Ellipse and the Lemniscate, the latter of which is employed here to 
illustrate the general problem. 

The moving pieces in a mechanism are rigid and inextensible and are always under certain 
conditions of restraint. " Conditions of restraint " may be illustrated by the familiar case of the con- 
necting-rod of the locomotive, one end of which is alwa3^s attached to the driving-wheel at the 
crank-pin and is therefore constrained to describe a circle about the axle of that wheel, while the 
other end of the rod must move in a straight line, being fastened by the "wrist-pin" to the "cross- 
head," which slides between straight " guides." The first step in tracing a point-path of anj^ 
mechanism is therefore the determination of the fixed points and a general analysis of the motion. 



==■ Locus is the Latin for place; and in father untechnical language, although in the exact sense in which it is used mathe- 
matically, we may say that the Locus of points or lines is the place where you \n-Ay expect to find them under their conditions 
of restriction. For example, the surface of a sphere is the locus of all points equidistant from a fixed point (its centre). The 
locus of a point moving in a plane so as to remain at a constant distance from a given lixed point, is a circle having the 
latter point as its centre. 



52 



THEORETICAL AND PRACTICAL GRAPHICS. 



158. We have given, in Fig. 98, two links or bars, MN and S P, fastened at N and P by- 
pivots to a third link, N P, while their other extremities are pivoted on stationary axes at M and 
S. The only movement possible to the point N is therefore in a circle about M; while P is 
equally limited to circular motion about S. The points on the link N^ P, with the exception of its 



2 MN _2 MS 
3 




THE LEMNISCATE 

AS A 

LINK-MOTION CURVE 

■extremities, have a compound motion, in curves whose form it is not easy to jaredict and which 
differ most curiously from each other. The figure-of-eight curve shown, otherwise the " Lemniscate 
of Bernoulli," is the point-path of Z, the link NP being supposed prolonged by an amount, P Z, 
■equal to N P. Since iVP is constant in length, if N were moved along to E the point P would 
have to be at a distance iV P irom E and also' on the circle to which it is confined ; therefore its 
new position, /, is at the intersection of the circle P s r by an arc of radius P N, centre F. Then 
Ff prolonged by an amount equal to itself gives /; , another point of the Lemniscate, and to which 
Z has then moved. All other positions are similarly found. 

If the motion of N is toward D it will soon reach a limit, A, to its fm-ther movement in that 
direction, arriving there at the instant that P reaches a, when NP and PS will be in one straight 
line, S ^1. In this position any movement of P either side of a will drag N back over its former 
path ; and unless P moves to the left, past a, it would also retrace its path. P reaches a similar 
■" dead point " at v. 

To get the curve illustrated, the links N P and P S had to be equal, as also the distance M S 
to M N. By varying the proportions of the links, the point-paths would be correspondingly affected. 



INSTANTANEOUS CENTR ES. — CENTRO IDS. 



53 



By tracing the path of a point on P N' produced, and as far from N as Z is from P, the 
student will ol>tain an interesting contrast to the Lemniscate. 

If M and S were joined liy a link and the latter held rigidly in position, it would have been 
called the fixed link; and although its use would not have altered the motions illustrated and it is 
not essential that it should be drawn, yet in considering a mechanism, as a whole, the line joining 
the fixed centres always exists, in the imagination, as a link of the complete system. 



INSTAXTAXEOUS CEXTEES. — CEXTROIDS. 



159. Let us imagine a l.ioy al>out to hurl a stone from a sling. Just before he releases it 
he runs forward a few steps, as if to add a little extra impetus to the stone. "While taking those few 
steps a peculiar shadow is cast on the road l)y the end of the sling, if the day is bright. The 



Dy 




boy moves with respect to the earth ; his hand moves in relation to himself, and the end of the 
sling describes a circle about his hand. The last is the only definite element of the three, j'et it 
is sufficient to simplify otherwise difficult constructions relating to the complex curve which is 
described relatively to the earth. 



54 T.HEORETICAL AND PRACTICAL GRAPHICS. 

A tangent and a normal to a circle are easily obtained, the former being, as need hardly be 
stated at this point, perpendicular to the radius at the point of tangency, while the normal simply 
coincides in direction with such radius. If the stone were released at any instant it would fly off 
in a straight line tangent to the circle it was describing about the hand as a centre; but such line 
would, at the instant of release, be tangent also to the compound curve. If then we wish a tangent 
at a given point of any curve generated by a point in motion, we have but to reduce that motion 
to circular motion about some moving centre; then joining the point of desired tangency with the — 
at that instant — position of the moving centre, we have the normal, a perpendicular to which gives 
the tangent desired. 

A centre which is thus used for an instant only is called an instantaneous centre- 
ISO. In Fig. 99 a series of instantaneous centres are shown and an important as well as inter- 
esting fact illustrated, viz., that every moving piece in a mechanism might be rigidly attached to a 
certain curve and by the rolling of the latter upon another curve, the link might be brought into 
all the positions which its visible modes of restraint compel it to take. 

161. In the "Fundamental" part of the figure AB is assumed to be one position of a link. We 
next find it, let us suppose, at A' £', A having moved over A A' and B over B B'. Bisecting 
A A' and B B' by perpendiculars intersecting at 0, and drawing A, A', OB, and OB', we 
have A A ' = 0^ = B B', and evidently a point about which, as a centre, the turning of A B 
through the angle ^i would have brought it to A' B'. Similarly, if 'the next position in which we 
find AB is A" B", we may find a point s as the centre about which it might have turned to 
bring it there; the angle being 0.,, probably different from 0^. 

The points n and m are analogous to and s. 

If s' be drawn equal to s and making with the latter an angle 0^ , equal to the angle 
A A', and if s were rigidly attached to A B the latter would be brought over to A' B' by 
bringing s' into coincidence with s. In the same manner, if we bring s' n' upon sn through 
an angle 0.^ about s, then the next position. A" B", would be reached by A B. 0' s' n' m' is then 
part of a polygon whose rolling upon s n m would bring A B into all the positions shown, provided 
the polygon and the line were so attached as to move as one piece. Polygons whose vertices are 
thus obtained are called central polygons. 

If consecutive centres were joined we would have curves, called centroids, instead of polygons ; 
the one corresponding to s n m being called the fixed, the other the rolling centroid. The perjDen- 
dicular from upon A A' is a normal to that path. But if A were to move- in a circular arc 
the normal to its path at any instant would be simply the radius to the position of A at that 
instant. 

If then both A and B were moving in circular paths we would find the instantaneous centre 
at the intersection of the normals (radii) to the points A and B. 

162. In Fig. 98 the instantaneous centre about which the whole link N P is turning is at the 
intersection of radii MN and SP (produced); and calling it A' we would have XZ normal at Z 
to the Lemniscate. 

163. The shaded portions of Fig. 99 illustrate some of the forms of centroids. 

The mechanism is of four links, opposite links equal. Unlike the usual quadrilateral fulfilling 
this condition, the long sides cross, hence the name "anti-parallelogram." 

The "fixed link (a)" corresponds to MS of Fig. 98 and its extremities are the centres of 
rotation of the short links, whose ends, / and /j, describe the dotted circles. 

For the given position T is evidently the instantaneous centre. Were a bar pivoted at T and 



TROCHOIDS. 55 

fastened at right angles to '" moving link (a)," an infinitesimal turning aljout T would move '' link 
(a) " exacth^ as under the old conditions. 

B}' taking "link (a)" in all possible positions and, for each, prolonging the radii through its 
extremities, the points of the fixed centroid are determined. Inverting the combination so that 
"moving link (a)" and its opposite are interchanged, and proceeding as before, gives the points of 
■"rolling centroid (a)." 

These centroids are branches of hyperbolas having the extremities of the long links as foci. 

By holding a short link stationary, as "fixed link (b)," an elliptical fixed centroid results; 
"rolling centroid (b) " being obtained, as before, by inversion. The foci are again the extremities of 
Ihe fixed and moving links. 

Obviously the curved pieces represented as screwed to the links would not be emploj^ed in a 
practical construction, and they are only introduced to give a more realistic effect to the figure and 
jjossibly thereby conduce to a clearer understanding of the subject. 

164. It is interesting to notice that the Lemniscate occurs here under new conditions, being 
"traced by the middle point of "moving link (a)." 

The study of kinematics is both fascinating and profitable, and it is hoped that this brief glance 
-at the subject may create a desire on the part of the student to pursue it further in such works 
as Reuleaux' Kinematics of Machinery and Burmester's Lehrbuch der Kinematik. 

Before leaving this topic the important fact should be stated, which now needs no argument to 
■establish, that the instantaneous centre for any position of the moving piece, is the point of contact 
of the rolling and fixed centroids. 

165. "We shall have occasion to use this principle in drawing tangents and normals to the 

TROCHOIDS 

which are the principal Roulettes, or roll-traced curves, and wliich may be defined as follows: — 

If in the same plane one of two circles rolls upon the other without sliding, the path of anj' 
point on the radius of the rolling circle or on the radius produced is a trochoid. 

166. The Cycloid. Since a straight line may be considered a circle of infinite radius the above 
definition would include the curve traced by a i^oint on the circumference of a locomotive wheel as 
it rolls along the rail, or of a carriage wheel on the road. This curve is known as a cycloid* and 
is shown in T n a b c, Fig. 100. It is the proper outline for a portion of each tooth in a certain 
case of gearing, viz., where one wheel has an infinite radius, that is, becomes a " rack." Were T^ 
3, ceiUng-corner of a room, and T,, the diagonally opposite floor-corner, a weight would sHde from 

Tg to Tj2 more quickly on guides curved in cycloidal shape than if shaped to any other curve or 
if straight. If started at c or any other point of the curve it would reach T,., as soon as if started 

at r, 

167. In beginning the construction of the cycloid we notice first that as T V D rolls on the 
straight fine A B the arrow D R T will be reversed in position (as at D-^ T^ as soon as the semi- 
circumference TSD has had rolling contact with A B. The tracing point will then be at T^, its 
maximum distance from A B. 

When the wheel has roUed itself out once ujion the rail the point T will again come in contact 
T\'ith the rail, as at T,,. 



* "Although the invention of the cycloid is iittributefl to Galileo, it is certain that the family of curves to which it belongs 
haa heen known and some of the properties of such curves investigated, nearly two thousand yeare before Galileo's time, if 
not earlier. For ancient astronomers explained the motion of the planets by supposing that each planet travels uniformly 
Tound a circle whose centre travels uniformly around another circle."— Proctor, Geometry of Cycloids. 



56 



THEORETICAL AND PRACTICAL GRAPHICS. 



The distance T T^^ evidently equals 2 tt r, when r = T R. 

If the semi-circumference To D (equal to ir r) be divided into any number of equal parts; and 
also the path of centres R R^ (again = 7rr) into the same number of equal parts, then as the points 
1, 2, etc., come in contact with the rail, the centre R will take the positions R^, R,, etc., directly 
above the corresponding points of contact. A sufficient rolling of the wheel to bring point 3 upon 
A B would evidently raise T from its original position to the former level of ~. But as T must always 
be at a radius' distance from R, and the latter would by that time be at R.^, we would find T 
located at the intersection (n) of the dotted line of level through 8 by an arc of radius R T, centre 
R^. Similarly for other points. 

The construction, summarized, iiivolves the drawing of lines of level through equidistant points of 
division on a semi-circumference of the rolling circle, and their intersection by arcs of constant radius 
(that of the rolling circle) from centres which are the successive positions taken by the centre of the 
rolling circle. 

It is worth while calling attention to a point occasionally overlooked by the novice, although 
almost self-evident, that in the position illustrated in the figure the point T drags behind the centre 
R until the latter reaches ii^, when it passes and goes ahead of it. From .R, the line of level 
through 5 could be cut not alone at c by an arc of radius c R, but also in a second point ; 
evidently but one of these points belongs to the cycloid, and the choice depends upon the direction 
of turning and the relative position of the rolling centre and the moving point. This matter requires 
more thought in drawing trochoidal curves in which both circles have finite radii, as will appear 
later. 

E'igr- loo- 

, THE CYCLOID. 




168. Were ^joints T^ and T^, given, and the semi-cycloid T^ T^.., desired, we can readily ascertain 
the " base," ^4 B, and generating circle, as follows : — -Join T|, with T,, ; at any point of such line, as 
X, erect a j^erpendicular, xyj. from the similar triangles x y T^, and T,. D.^ T^^ having angle <t> common 
and angles 6 equal we see that 

xy:xT,,::T,D,:D,T,,::2r:7rr::2:7r::l:~ov, very nearly, as U : 22. 

If, then, we lay off x Tj^ equal to twenty-tioo equal parts on any scale, and a perpendicular, x y, 
fourteen parts of the same scale, the line y T-^.^ will be the base of the desired curve; while the di- 
ameter of the generating circle will be the perpendicular from Tg to y T^.^ prolonged. 

169. To draw a tangent to a cycloid at any point is a simple matter, if we see the analogy 
between the 'point of contact of the wheel and rail at any instant, and the hand used in the former 
illustration (Art. 159). At any one moment each point on the entire wheel may be considered as 
describing an infinitesimal arc of a circle whose radius is the line joining the point with the point 
of contact on the rail. The tangent at N, for examjDle, (Fig. 100), would be t N, i^erpendicular 
to the normal. No, joining N with o; the latter point being found by using iV as a centre and 



THE CYCLOID. — COMPANION TO THE CYCLOID. 



57 



cutting A B by an arc of radius equal to m I, in which m is a point at the level of N on any 
position of the rolling circle, while I is the corresponding point of contact. The point o might also 
have been located by the following method: Cut the hne of centres by an arc, centre N, radius 
T R; would obviously be vertically below the position of the rolling centre thus determined. 

170. Till' Companion to the Cycloid. The kinematic method of drawing tangents, just applied, was 
devised by Roberval, as also the curve named by him the " Companion to the Cycloid," to which 
allusion has already been made (Art. 120) and which was invented by him in 1634 for the purpose 
of solving a problem upon which he had spent six years without success and which had foiled 
Galileo, viz., the calculating of the area between a cycloid and its base. Galileo was reduced to the 
expedient of comjoaring the area of the cj'cloid with that of the rolhng circle by weighing paper 
models of the two figures. He concluded that the area in cj[uestion was nearly but not exactly 
three times that of the rolling circle. That the latter would have been the correct solution may be 
readily shown by means of the " Comi^anion," as will be found demonstrated in Art. 172. 

171. Suppose two points coincident at T, (Fig. 101), and starting simultaneously to generate curvesi 
the first of these points to trace the cycloid during the rolling of circle T V D while the second is to 
move independently of the circle and so as to be always at the level of the point tracing the cycloid, 
yet at the same time vertically above the point of contact of the circle and base. This makes the 
second point always as far from the initial vertical diameter, or axis, of the cycloid as the length 
of the arc from T to whatever level the tracing point of the latter has then reached; that is, MA 
equals arc THs; RO equals quadrant Tsy. 

Adopting the method of Analytical Geometrj^, by using as a reference point, or origin, 
we may reach any point, ^-l, on the cur^'e, by co-ordinates, as x, x A. of which the horizontal is 
called an abscissa, the vertical an ordinate. By the preceding construction Ox equals arc sfy, while 
X A equals s tv — the sine of the same arc. The " Companion to the Cycloid " is therefore a curve 
of sines or sinusoid, since starting from the abscissas are ecjual to or proportional to the arc of a 
circle while the ordinates are the sines of those arcs. 

This curve is particularlj^ interesting as " expressing the law of the vibration of perfectly ■ elastic 
sohds ; of the vibratory movement of a particle acted upon by a force which varies directly as the 
distance from the origin ; apiDroximately, the vibratory movement of a pendulum ; and exactly the 
law of vibration of the so-called mathematical pendulum."* 

172. From the symmetry of the 
sinusoid with respect to R R^ and to 
we have area TAO R= E R,; 
adding area D E L R to both mem- 
bers we have the area between the 
sinusoid and T D and D E equal to 
the rectangle R E, or one-half the rect- 
angle D E K T ; or to -, ir r x 3 r = 

IT r ^, the area of the rolling circle. 

As T A C E is but half of the entire sinusoid it is evident that the total area below the curve 
is twice that of the generating circle. 

The area between the cycloid and its " companion " remains to be determined, but is readily 
ascertained by noting that as any jjoint of the latter, as A, is on the vertical diameter of the circle 



^-igr- loi. 




* Wood. Elements of Co-ordinate Geometry, p. 209. 



58 



THEORETICAL AND PRACTICAL GRAPHICS. 



passing through the then position of the tracing point, as a, the distance, A a, between the two 
curves at any level, is merely the semi-chord of the rolhng circle at that level. But this, evidently, 
equals Ms, the semi-chord at the same level on the equal circle. The equality of Ms and A a 
makes the elementary • rectangles MsSim^ and A A^ a^ a equal; and considering all the possible 
similarly constructed rectangles of infinitesimal altitude, the sum of those on semi-chords of the 
rolling circle would equal the area of the semi-circle TDy, which is therefore the extent of the area 
between the two curves under consideration. 

The figure showing but half of a cycloid, the total area between it and its "companion" must 
be that of the rolling circle. Adding this to the area between the " companion " and the base 
makes the total area between cycloid and base equal to three times that of the rolling circle. 

173. The paths of points carried by and in the plane of the rolUng circle, though not on its 
circumference, are obtained in a manner closel)^ analogous to that employed for the cycloid. 

In Fig. 102 the looped curve, traced by the arrow-point while the circle GHM rolls on the 

base A B, is called the Curtate Trochoid. To obtain the various positions of the tracing point T 

describe a circle through it from centre R. On this circle lay off any even number of equal arcs and 

draw radii from R to the points of division; also "lines of level" through the latter. The radii 

drawn intercept equal arcs on the rolling circle C H M. While the first of these arcs rolls upon 

AB the point T turns through the angle TRl about R and reaches the line of level through 

point 1. But T is always at the distance R T (called the tracing raclm) from R; and, as R has 

reached R^ in the rolling supposed, we find T^ — the new position of T — by an arc from R-^, radius 

T R, cutting said line of level. 

rE-ig-- loa. 
2 

1^ ' ^ 




After what has preceded, the figure may be assumed to be self-interpreting, each position of T 
being joined with the position of R which determined it. 

174. Were a tangent wanted at any point, as T^, we have, as before, to determine the point of 
contact of rolling circle and line when T reached T,, and use it as an instantaneous centre. Tj 
was obtained from i?, ; and the point of contact must have been vertically below the latter and on 
A B. Joining such jjoint to T, gives the normal, from which the tangent follows in the usual way. 

17.5. The Prolate Trochoid. Had we taken a point inside^ of the circle C H M and constructed its 
path the only difference between it and the curve illustrated would have been in the name and the 



HYPO-, EPI- AND PERI-TROCHOIDS. 



59 



shape of the curve. An undulating, wavy path would have resulted, called the prolate trochoid; but, 
as before, we would have described a circle through the tracing point; divided it into equal parts; 
drawn lines of level, and cut them by arcs of constant radius, using as centres the successive 
positions of R. 

HYPO-, EPI- AND PERI-TROCHOIDS. 

176. Circles of finite radius can evidently be tangent in but two ways — either externally, or 
internally; if the latter, the larger may roll on the one within it, or the smaller may roll inside 
the larger. When a small circle rolls within a larger the radius of the latter may be greater than 
the diameter of the rolling circle, or may equal it, or be smaller. On account of an interesting 
property of the curves traced by points in the jilanes of such rolling circles, viz., their capability of 
being generated, trochoidally, in two ways, a nomenclature was necessarj' which should indicate how 
each curve was obtained. This is included in the tabular arrangement of names below and which 
was the outcome of an investigation made by the writer in 1887 and presented before the American 
Association for the Advancement of Science.* In acceptmg the new terms advanced at that time 
Prof. Francis Reuleaux suggested the names Ortho-cycloids and Oyclo-orthoids for the classes of curves 
of which the cycloid and involute are respectively representative; orthoids being the paths of points 
in a fixed position with resi^ect to a straight line rolling ujion any curve, and cyclo-orthoid therefore 
implying a circular director or base-curve. These appropriate terms have been incorporated in the 
table. 

For the last column a point is considered as ivithin the rolling circle of infinite radius when on 
the normal to its mitial position and on the side toward the centre of the fixed circle. 

As will be seen bj^ reference to the Appendix, the curves whose names are preceded by the same 
letter may be identical. Hence the terms curtate and prolate, while indicating whether the tracing 
point is beyond or within the circumference of the rolling circle, give no hint aa to the actual form 
of the curves. 

In the table R represents the radius of the rolling circle, F that of the fixed circle. 

NOMENCLATUKE OF TROCHOIDS. 



Position of 
Tracing 


Circle rolling 

upon 
Straight Line. 


Circle rolling upon circle. 


Straight Line 

rolling upon 

Circle. li = a> 


External 
contacL- 


Internal contact. 


or 




Epitrochoids. 


Larger Circle 
rolling. 


Smaller circle rolling. : 


Cyclo- 
orthoids. 


Describing j 

„ . Oi-tho-cycloias. 
Point. 1 


2 R > F. 1 2 K < F. 1 2 R = P. 


Perltrochoids. 


Major 
Hypotrochoids. 


Minor 
Hypotrochoids. 


Medial 
Hypotrochoids. 1 


On circumference i rvelold 
of rolling circle. v-jv, uiu. 


(a) Epicycloid. 


(a) Pericyclold 


(d) Jlajor 

1 Hypocyclolds. 


(d) Minor 
Hypocycloid. 


Straight j 
Hypocycloid. 


Involute. 


Within Prolate 
Circnmference. Troehoid. 


(b) Prolate 
Epitrochoid. 


(c) Prolate 
Perltrochold. 


(e) Jlajor Prolate (f) Jlinor Prolate '^' Eu'im^cal ' 
Hypotrochoid. i Hypotrochoid. Hypotrochoid. 


Prolate 
Cyclo-orthoid. 


Without 
Circumference. 


Curtate 1(c) Curtate (b) Curtate 1 ^^ curSe '" Cumt'e *°^ E^ilpttcll 
Trochoid. ;j Epitrochoid. ' Perltrochold. |j HypotiSJhold. Hypotrochoid. Hypot^ochiid. 


Curtate 
Cyclo-orthoid. 



177. From the above we see that the prefix qn (over or upon) denotes the curves resulting 
from external contact; %j50 (under) those of internal contact with smaller circle rolUng; while 
peri (about) indicates the third possibility as to rolling. 



* Ke-prlnted in substance in the Appendix. 



60 



THEORETICAL AND PRACTICAL GRAPHICS. 



:F'ig-- 1.03. 




178. The construction of these curves is in closest analogy to that of the cycloid. If, for 
example, we desire a rtiajor hypocydoid we first draw two circles, mVP. m x L, (Fig. 103), tangent 

internally, of which the rolling circle has its 
diameter greater than the radius of the fixed 
circle. Then, as for the cycloid, if the tracing- 
point is P, we divide the semi-circumference m V P 
into equal parts and from the fixed centre, F, de- 
scribe circles through the jjoints of division, as 
those through 1, 2, 5, ^ and 5. These replace the 
"lines of level" of the cycloid, and may be called 
circles of distance, as they show the distance from 
F of the point P, for definite amounts of angular 
rotation of the latter. For, if the circle P Vm 
were simply to rotate about R, the point P would 
reach m during a semi-rotation and would then 
be at its maximum distance from F. After turning 
through the equal arcs P 1, 1-2, etc., its distances 
from F would be i^a_and Fh respectively. 

If, however, the turning of P about R is due 
to the rolling of circle P Vm upon the arc m x z 
then the actual position of P, for any amount of turning about R, is determined by noting the new 
position of R, due to such rolling, as R^, R^, etc., and from it as a centre cutting the proper circle 
of distance by an arc of radius R P. 

Since the radius of the smaller circle is in this case three-fourths that of the larger, the angle 
niFz (135°) at the centre of the latter intercepts an arc, m x z, equal to the 180° arc, m V P, on the 
smaller circle. Equal arcs on imequal circles are subtended by angles at the centre which are inversely 
proportioned to the radii. 

While arc mVP rolls upon arc mxz the centre R will evidently move over circular arc 
R — iig . Divide mxz into as many equal parts as m V P and draw radii from F to the points of 
division ; these cut the jDath of centres at the successive positions of R. When arc m 5-Jf, for 
example, has rolled upon its equal m u v then R will have reached R^ ; P will have turned about 
R through angle PR2 = mR4 and will be at n, the intersection of hfg — the circle of .distance through 
2 — by an arc, centre R,, radius R P. Similarly for other points. 

179. To trace the path of any point on the circumference of a circle so rolling as to give the 
epi- or pen-cycloid requires a construction similar at every step to that just illustrated. The same 
remark ajsplies equally to the determination of the paths of points within or beyond the circum- 
ference of the rolling circle, as will be seen by reference to Fig. 104, in which the path of the 
point P is determined (a) as carried by the circle called "first generator," rolling on the exterior 
of the " first director " ; (b) as carried by the " second generator " which rolls on the exterior of the 
"second director" — which it also encloses. In the first case the resulting curve is a prolate epi- 
trochoid; in the second a curtate pjeritrochoid. Proceeding in the usual way, a semi-circle is drawn 
through P from each rolling centre, R and p. Dividing these semi-circles into the same number of 
equal jjarts draw next the dotted " circles of distance " through these points, all from centre F. 
The figure illustrates the sjDecial case where the larger set of " circles of distance " divides both serai- 
circumferences into equal parts. The successive positions of P, as P^, P.^, etc., are then located by 



EPI- AND PERI-TROCHOIDS. 



61 



arcs of radii R P or p P, struck from the successive jjositions of R or p and intersecting the proper 
" circle of distance." For example, the turning of P through the angle P R 1 about R would bring 
P somewhere upon the circle of distance through point 1; but that amount of turning would be 
due to the rolling of the first generator over the arc m Q, which would carry R to R^; P would 







therefore be at Pj , at a distance R P from R^ and on the dotted arc through 1. Similarly in 
relation to p. Each position of P is joined with each of the centres from which it could be 
obtained. 



62 



THEORETICAL AND PRACTICAL GRAPHICS. 




SPECIAL TROCHOIDS. 

180. The Ellipse and Straight Line. Two circles are called Cardanic* if tangent internally and 
the diameter of one is twice that of the other. If the smaller roll in the larger all points in the 
plane of the generator will describe ellipses except points on the circumference, each of which will 
move in a straight line — a diameter of the director. Upon this latter property the mechanism known 
as " White's Parallel Motion " is based, in which a piston-rod or other piece intended to have 
reciprocating rectilinear motion is pivoted to a small gear-wheel or i^inion which rolls on the interior 
of a toothed annular wheel of twice the diameter of the pinion. 

181. The Limacon and Cardioid. The Limafon is a curve whose points may be obtained by 
^ig-- i-os. drawing random secants through a point on the circumference of a 

circle and on each laying off a constant distance, on each side 
of the second point in which the secant cuts the circle. 

In Fig. 105 let v and d be random secants of the circle 
0ns; then if n v, n p), c a and c d are each equal to some con- 
stant, b, we shall have v, p, a and d as four points of a Limagon. 
Referring any point as d to and the diameter s, we have 
d = p = c + c d = 3 r cos + h, while a = 2 r cos 6 — 6, 
whence the general polar equation for the Limagon, p = 2 r cos 
e + h. 
When h ■= 2 r the Limagon becomes the heart-shaped curve called the Cardioid."^ 

182. All Limagons, general and special, may be generated either as epi- or peri-trochoidal 
curves : as epi-trochoids the generator and director must have equal diameters, any point mi the 
circumference of the generator then tracing a Cardioid, while anj^ point on the radius (or radius 
produced) describes a Limagon; as peri-trochoids the larger of a pair of Cardanic circles must roll 
■on the , smaller, the Cardioid and Limagon then resulting, as before, from the motion of jDoints 
respectively on the circumference of the generator or within or ivithout it. 

183. In Fig. 106 the Cardioid is obtained as an epicycloid, being traced by point P during one 
revolution of the generator P H m about an equal directing circle m s 0. 

As a Limagon we may get points of the Cardioid, as y and z, by drawing a secant through 
and laying off s y and s z each equal to 2 r. 

184. The Limacon as a Trisectrix. Three famous problems of the ancients were the squaring of 
the circle, the duplication of the cube and the trisection of an angle. Among the interesting curves 
invented by early mathematicians for the purjaose of solving the latter problem were the Quadratrix 
and Conchoid, whose construction is given later in this chapter ; but it has been found that certain 
trochoids also possess this interesting property, among them the Limagon of Fig. 106, frequently 
called the Epitrochoidal Trisectrix. Its construction as an epitrochoid need not be described in detail 
after what has preceded. 

As a Limagon we would find points as G and X by drawing from R a secant RX io the circle 
■called " path of centres " and making S X and S G each equal to the radius of that circle. 

185. To trisect an angle, as M R F, bisect one side of the angle as RF at m; use m R and 
m F as radii for generator and director resijectively of an epitrochoid having a tracing radius, R F, 
■equal to twice that of the generator. Make R N = R F and draw NF; this will cut the Limagon 



*Tenn due to Keuleaux and based upon tlie fact that Cardano (16th century) was probably the fir^t to investigate the 
paths described by points during their rolling, 
t From CardU, the Latin for heart. 



SPECIAL TROCHOIDS. 



63 



F T^R Q (traced by point F as carried bj' the given generator) in a point T^ . The angle T^ R F 
will then be one-third of N R F, which may be proved as follows: F reaches T^ by the rolling of 
arc m n on arc m 1\ . These arcs are subtended bj^ equal angles, 4> , the circles being equal. During 
this rolling R reaches R^ , bringing i? i^ to R^ T^. In the triangles Ti R^ F and R F R^ the side 
F Ri is common, angles <^ equal and side R^ T^ equal to side R F; the line R T^ is therefore 
parallel to Ri F, whence angle T^ R F must also equal <^. In the triangle R F R^ we denote by 6 

180° — <iy 



the angles opposite the equal sides RF and R^F; then J B + 4>^ 1S0° or 6 = 



2 



In triangle 



N R F we have the angle at F equal to 6 — <^, and J {6 — <^) + .r + <^ = 180° which gives x ^= 2 <j> 
by substituting value of 6 from previous equation. 



x'lg-. loe. 



FOT_Cardioid 




186. The Involute. As the opposite extreme of a circle rolling on a straight line we may have 
the latter rolHng on a circle. In this case the rolling circle has an infinite radius. A point on the 
straight line describes a curve called the involute. This would be the path of the end of a thread 
if the latter were in tension while being unwound irom a spool. 

In Fig. 107 a rule is shown, tangent at u to a circle on which it is supposed to roll. Were a 
pencil-point inserted in the centre of the circle at j (which is on line m x produced) it would trace 
the involute. When j reaches a the rule will have had rolling contact with the base circle over an 
arc uts — a whose length equals Une u x j. Were a the initial point we may obtain b, c, etc., bj' 
making tangent m h = arc in a ; tangent n c ^ arc n a, etc. Each tangent thus equals the arc from 
initial point to i^oint of tangency. 



64 



THEORETICAL AND PRACTICAL GRAPHICS. 



187. The circle from which the invokite is derived or evolved is called the evolute. Were a 
hexagon or other figure to be taken as an evolute a corresponding involute could be derived ; but 
the name " involute," unqualified, is understood to be that obtained from a circle. 

From the law of formation of the involute the rolling line is in all its positions a normal to 
the curve; the point of tangencj' on the evolute is an instantaneous centre, and a tangent at any 
point as / is a jDerpendicular to the tangent, f q, fi-om / to the base circle. 

Like the cycloid, the involute is a correct working outline for the teeth of gear-wheels ; and 
gears manufactured on the involute system are to a considerable degree supplanting other forms. 

A surface known as the developable helicoid is formed by moving a line so as to be always 
tangent to a given helix. It is interesting in this connection to notice that any plane perpendicular 
to the axis of the helix would cut such a surface in a pair of involutes.* 




188. The Spiral of Archimedes. This is the curve that would be generated by a point having a 
uniform motion around a fixed jDoint — the pole — combined with uniform motion toward or fr'om the 
pole. 

In Fig. 107,. with as the pole, if the angles are ec[ual and D, E and y^ are in arith- 
metical progression then the points D, E and y^ are points of an Archimedean Spiral. 

This spiral can be trochoidally generated, simultaneously with the involute, by inserting a pencil 
point at 2/ in a piece carried by — and at right angles with — the rule, the point y being at a distance, 
xy, from the contact-edge of the rule, equal to the radius Os of the base circle of the involute; for 



* The day of writing tlie above article the following item aijpeared in the New York Eoening Post: "Visitors to the Royal 
Observatory, Greenwich, will hereafter miss the great cylindrical structure which has for a quarter century and more covered 
the largest telescope possessed by the Observatory. Notwithstanding its size the Astronomer Boj'al has now procured through 
the Lords Commissioners a telescope more than twice as large as the old one. . . . The optical peculiarities embodied in the 
new instrument will render it one of the three most powerful telescopes at present in existence. . . . The peculiar architectural 
feature of the building which is to shelter the new telescope is that its dome, of thirty-six feet diameter, will surmount a 
tower having a diameter of only thirty-one feet. Technically the form adopted is the surface generated by the revolution of 
an involute of a circle.^^ 



SPECIAL TROCHOIDS. 



65 



after the rolling of u x over an are ii t we shall have t x, as the portion of the rolling line between 
X and the point of tangency, and x y will have reached x, y^. If tlie rolling be continued y will 
evidently reach 0. We see that y = u x and y^ = t x^; but the lengths u x and t x, are propor- 
tional to the angular movement of the rolling line about 0, and as the spiral may be defined as 
that curve in which the length of a radius vector is directly proi^ortional to the angle through which 
it has turned about the pole, the various positions of y are evidently points of such a curve. 

189. Were the pole, 0, given and a portion only of the spiral, we could draw a tangent at 
any point, y^, by determining the circle on which the spiral could be trochoidally generated, then 
the instantaneous centre for the given position of the tracing-point, whence the normal and tangent 
would be derived in the usual way. The radius Ot of the base circle would equal lo y, — the 
difference between two radii vectores Oy and Oz which include an angle of -57? 29 +, (the angle 
which at the centre of a circle subtends an arc equal to the radius). The instantaneous centre, t, 
would be the extrenaity of that radius which was perpendicular to y^. The normal would be 
t i/i and the tangent T 1\ perpendicular to it. 

190. This spiral is the proper outline for a cam to convert uniform rotary into uniform recti- 
linear motion, and when comljined with an equal and opposite spiral gives the well known form 
called the heart-cam. As usually constructed the acting curve is not the true spiral but a curve 
whose jioints are at a constant distance from 'J---- 'e'^s- a-cs. 
the theoretical outline equal to the radius of 
the friction-roller which is on the end of the 
piece to lie raised. A small portion of such 
a " parallel curve " is indicated in the upper 
part of Fig. 107. 

191. If a point travel on the surface of 
a cone so as to combine a uniform motion 
around the axis with a uniform motion toward 
the vertex it will trace a conical helix whose 
orthograjDhic projection on the plane of the 
base will be a Spiral of Archimedes. 

In Fig. 108 a to]:) and fi'ont view are 
given of a cone and helix. The shaded por- 
tion is the development of the cone, that is, 
the area equal to the convex surface and 
which — if rolled up — would form the cone. 
To obtain the development draw an arc 
A' G" A" of radius equal to an element. The 
convex surface of the cone will then be repre- 
sented by the sector A' 0' A" whose angle 6 
may be found b}' the proportion A : 0' A' : : 
6:360°, since the arc A' G" A" must equal 
the entire circumference of the cone's base. 

The student can make a paper model of 
the cone and helix by cutting out a sector of 
a circle, making allowance for an overlap on which to put the mucilage, as shown by the dotted 
lines 0' y and yvz in the figure. 




66 THEORETICAL AND PRACTICAL GRAPHICS. 

The development of the conical helix is evidently the same kind of spiral as its orthographic 
projection. 

PARALLEL CURVES. 

192. A parallel curve is one whose points are at a constant normal distance from some other 
curve. Parallel curves have not the same mathematical properties as those from which they are 
derived, except in the case of a circle ; this can readily be seen from the cam fig-ure under the last 
heading, in which a point, as S-^, of the true spiral, is located on a line from which is by no 
means in the direction of the normal to the curve at S, , upon which lies the jjoint S^ of the 
parallel curve. 

E-ig-. lOS. 




Instead of actually determining the normals to a curve and on each laying off a constant 
distance we may draw many arcs of constant radius, having their centres on the original curve; the 
desired parallel will be tangent to all these arcs. 

In strictly mathematical language the parallel curve is the envelope of a circle of constant radius 
whose centre is on the original curve. We may also define it as the locus of consecutive inter- 
sections of a system of equal circles having their centres on the original curve. 

If on the convex side of the original the parallel will resemble it in form, but if loithin the 
two may be totally dissimilar. This is well illustrated by Fig. 109 in which the parallel to a 
Lemniscate is shown. 

The student will obtain some interesting results by constructing the parallels to ellipses, parabolas 
and other plane curves. 

THE CONCHOID OF NICOMEDES. 

193. The Conchoid, named after the Greek word for shell," may be obtained by laying off a constant 
length on each side of a given line M N (the directrix) upon radials through a fixed point or 
pole, (Fig. 110). If m v = m n = s x then v, n and x are points of the curve. Denote by 
a the distance of fi-om M A^ and use c for the constant length to be laid off; then if a < c 
there will be a loop in that branch of the curve which is nearest the pole, — the inferior branch. 
If a = c the curve has a point or cusp at the pole. When a> c the curve has an undulation or 
wave -form towards the jDole. 

* A series of curves much more closely resembling those of a shell can be obtained by tracing the paths of points on the 
piston-rod of an oscillating cylinder. See Arts. 157 and 158. 



THE CONCHOID.— THE QUADRATRIX. 



67 



V = c + Om; On= c — Oin; we may therefore express the relation to of points on the 
curve by the equation p = c ± i7i = c ± « sec <f>. 



X'ig-- llO- 




194. Mention has ah-eady been made of the fact that this was one of the curves invented for 
the purpose of solving the problem of the trisection of an angle. ^^'ere the angle m x (or ^) we 
would first draw p qr, the superior branch of a conchoid having the constant, c, equal to twice 
Dm. A parallel from m to the axis will intersect the curve at q; the angle p Oq will then be 
one-third of <^: for since 6 f/ = -.' m we have m q ^ 2 m cos /3 ; also m q : m : : sin : sin fi (the 
sides of a triangle being proportional to the sines of the ojiposite angles) ; therefore 8 m cos (3 : 
m : : sin 6 : sin ji, whence sin d ^ " sin ft cos /3 = sin 2 ft (fi-om known trigonometric relations). The 
angle ^ is therefore equal to twice y3, making the latter one-third of angle 4>- 

195. To draw a tangent and normal at any point v we find the instantaneous centre o on the 
principle that it is at the intersection of normals to the paths of two moving points of a line, the 
distance between said points remaining constant. The motion of c in tracing the curve is — at the 
instant considered — in the direction Ov; Oo is therefore the normal. The point m of Ov is at 
the same moment moving along M N, for which mo is the normal. Their intersection o is then 
the instantaneous centre and o v the normal to the conchoid, with v z perpendicular to o v for the 
desired tangent. 

196. This interesting curve may be obtained as a plane section of one of the higher mathematical 
surfaces. If two non-intersecting lines — one vertical, the other horizontal — be taken as guiding lines 
or directrices of the motion of a third straight line whose inclination to a horizontal plane is to be 
constant, then every horizontal plane will cut conchoids from the surface thus generated, while every 
plane parallel to the directrices will cut hyperbolas. From the nature of its j^'ane sections this 
surface is called the Conchoidal Hyperboloid. 



THE QUADR.XTRIX OF DINOSTEATl'S. 

197. In Fig. Ill let the radius T rotate uniformly about the centre; simultaneously with its 
movement let M N have a uniform motion parallel to itself, reaching .1 B at the same time with 
radius OT; the locus of the intersection of .1/ .V witli the radius will he the Quadratriz. Points 



68 



THEORETICAL AND PRACTICAL GRAPHICS. 



^ E'igr- 111- 



,/z 



exterior to the circle may be found by prolonging the radii while moving If iV away from A B. 
As the intersection of MN with OB is at infinit}^ the former becomes an asymptote to the curve 
as often as it moves from the centre an additional amount equal to the diameter of the circle; 
the number of branches of the Quadratrix may therefore 
be infinite. It may be proved analytically that the curve 
crosses A at a distance from equal to 3 r -r- ir. 

198. To trisect an angle, as T a, by means of the 
Quadratrix draw the ordinate ap, trisect p T hj s and x 
and draw sc and xm; radii Oc and Om will then 
divide the angle as desired: for by the conditions of 
generation of the curve the line MN takes three equi- 
distant parallel positions while the radius describes three 
equal angles. 

THE CISSOID OF DIOCLES. 

199. This curve was devised for the purpose of obtaining two mean proportionals between two 
given quantities, by means of which one of the great problems of the Greek geometers — the dupli- 
cation of the cube — might be effected. 

The name was suggested by the Greek word for ivy since " the curve appears to mount along 
its asymptote in the same manner as that parasite plant climbs on the tall trunk of the pine."* 

This was one of the first curves invented after the discovery of the conic sections. Let C (Fig. 
112) be the centre of a circle, A C E a, right angle, NS and 31 T any pair of ordinates parallel to 

i^ig-. iia. 





asynljitote 



and equidistant from CE; then either secant from A through the extremity of one ordinate will 
meet the other ordinate in a point of the cissoid; P and Q, for example, will be points of the 
curve. 

The tangent to the circle at B will be an asymptote to the curve. 

It is a somewhat interesting coincidence that the area between the cissoid and its asymptote is 
the same as that between a cycloid and its base, viz., three times that of the circle from which 
it is derived. 



* Leslie. Geometrical Analysis. 1821. 



THE GISSOID. — THE TRACTRIX. 69 

200. Sir Isaac Newton devised the following method of obtaining a cissoid by continuous motion : 
Make AV=AC; then move a right-angled triangle, of base = V C, so that the vertex F travels along 
the hne DE while the edge JK always passes through V; then the middle point, L, of the base FJ, 
will trace a cissoid. This construction enables us readily to get the instantaneous centre and a tangent 
and normal; for Fn is normal to FC — the path of F, while nz is normal to the motion of J toward 
/ V ; the instantaneous centre n is therefore at the intersection of these normals. For any other 
point as P we apply the same principle thus: With radius equal io A C and from centre P obtain 
x; draw P x, then Vz p)arallel to it; a vertical from x will meet Vz at the instantaneous centre j/,. 
trom which the normal and tangent result in the usual way. 

201. Two quantities in and n will be mean proportionals between two other quantities a and b 
if m^ ^ n a and n"' = m b ; that is, if m' ^ a'' b and if n" = a b''. 

If 6 = i? a we will have m for the edge of a cube whose volume will be twice that of a^, when 
m' = a'"' b. 

To get two mean proportionals between quantities r and h make the smaller, r, the radius of 
a circle from which derive a cissoid. Were APR the derived curve we would then make Ct equal 
to the second quantity, b, and draw B t, cutting the cissoid at Q. A line A Q would cut off on 
C t a distance C v equal to m, one of the desired proportionals ; for vi' will then equal r- b, as maj^ 
be thus shown by means of similar triangles : — 

Cv: MQ:: CA:MA whence C v' = - '^^^ (1) 

Of. MQ::CB:BM " '^^=''^ (2) 

M Q ■ 31 A -.-.SNiAN:: VAN. BN: A N, whence MQ= ^^^^^^^-^N . _ (3) 

From (2) we have M Q = ^^(^^=^) (4) 

" (3) " " MQ^-=^^^^^§^^^ (5) 

Keplacing M Q^ in equation (1) by the product of the second members of equations (4) and (5) 
gives Cv'' (i. e., m^) = r- b. 

By interchanging ;■ and b we obtain n, the other mean projDortional ; or it might be obtained 
by constructing similar triangles having a, b and m. for sides. 

THE TEACTEIX. 

202. The Tractrix is the involute of the curve called the Catenary (later described) yet its usual 
construction is based on the fact that if a series of tangents be drawn to the curve the portions 
of such tangents between the points of tangency and a given line will be of the same length; or, in 
other words, the intercept on the tangent between the directrix and the curve will be constant. A 
practical and very close approximation to the theoretical curve is obtained by taking a radius Q R 
(Fig. 113) and with a centre a, a short distance from R on Q R, obtaining 6 which is then joined 
with a. On a b a centre c is similarly taken, whence c d is obtained. A sufficient repetition of 
this process will indicate the curve by its enveloping tangents; or a curve may actually be drawn 
tangent to all these lines. Could we take a, b, c, etc., as mathematically comecutire points the curve 
would be theoretically exact. 

The line Q S is an asymptote to the curve. 



70 



THEORETICAL AND PRACTICAL GRAPHICS. 




^ig-. ll-i. 



The area between the completed branch R P S and the hnes Q R and Q S would be equal to 
a quadrant of the circle on radius Q R. 

M S =^R 203. The surface generated by revolving the trac- 

trix about its asymptote has been employed for the 
foot of a vertical spindle or shaft and is known as 
Schiele's Anti-Friction Pivot. The step for such a pivot 
is shown in sectional view in the left half of the figure. 
Theoretically the amount of work done in overcoming 
friction is the same on all equal areas of this surface. 
In the case of a bearing of the usual kind, for a 
cylindrical spindle, although the pressure on each square 
inch of surface would be constant yet as unit areas at 
different distances from the centre would pass over very 
different amounts of space in one revolution, the wear 
upon them would be necessarily unequal. The rationale of 
the tractrix form will become evident from the following 
^ig-- 113- consideration: — If about to split a log, and having a 

choice of wedges, any boy would choose a thin one rather than one with a large angle, although 
he might not be able to prove by graphical statics the exact amount of advantage the one would 
have over the other. The theory is very simple, how- 
ever, and the student may profitably be introduced to it. 
SujDpose a ball, c, (Fig. 114) struck at the same instant 
by two others, a and b, moving at rates of six and eight 
feet a second respectively. On a c and b c prolonged take 
c e and c h equal, respectively, to six and eight units of 
some scale; complete the parallelogram having these lines 
as sides; then it is a well known principle in mechanics* that cd — the diagonal of this parallel- 
ogram — will not only represent the direction in which the ball c will move but also the distance — 
in feet, to the scale chosen — it will travel in one second. Obviously, then, to balance the effect of 
balls a and 6 upon c, a fourth would be necessary, moving from d toward c and traversing d c in 
the same second, that a and b travel, so that impact of all would occur simultaneously. These 
forces would be represented in direction and magnitude (to some scale) by the .shaded triangle 
c' d' c', which illustrates the very important theorem that if the three sides of a triangle — taken like 
c' e', e' d', d' c', in such order as to bring one back to the initial vertex mentioned — represent in 
magnitude and direction three forces acting on one point, then these forces are balanced. 

Constructing now a triangle of forces for a broad and thin 
wedge, (Fig. 115) and denoting the force of the supposed equal blows 
by F in each triangle, we see that the pressures are greater for the 
thin wedge than for the other; that is, the less the inclination to 
the vertical the greater the pressure. A pivot so shaped that as 
the pressure between it and its step increased the area to be traversed 
diminished would therefore, theoretically, be the ideal; and the rate of 
'tS~o-~~J change of curvature of the tractrix as its generating jDoint approaches 




IFig-. lis. 




the axis makes it, obviousl}', the correct form. 



*Foi- a demonstration llie student may refer to Rankine's Applied Mechanics, Art. 51. 



THE TRAGTRIX.— WITCH OF A G NISI. — CARTESIAN OVALS. 



71 



204. Navigator's charts are usually made by Mercator^s projection (so-called, not being a projection 
in the ordinary sense, liut with the extended signification alluded to in the remark in Art. 2). 
Maps thus constructed have this advantageous feature, that rhumb lines or loxodromics — the curves on 
a sphere that cut all meridians at the same angle — are represented as straight lines, which can only 
be the case if the meridians are indicated by parallel lines. The law of convergence of meridians 
on a sphere is, that the length of a degree of longitude at any latitude equals that of a degree on 
the equator multiplied by the cosine (see footnote, p. 31,) of the latitude; when the meridians are 
made non-convergent it is, therefore, manifestlj^ necessary that the distance apart of originally equi- 
distant parallels of latitude must increase at the same rate; or, otherwise stated, as on Mercator's 
chart degrees of longitude are all made equal, regardless of the latitude, the constant length repre- 
sentative of such degree bears a varying ratio to the actual arc on the sphere, being greater with 
the increase in latitude; but the greater the latitude the less its cosine or the greater its secant; 
hence lengths representative of degrees of latitude will increase with the secant of the latitude. 
Tables have been constructed giving the increments of the secant for each minute of latitude; but 
it is an interesting fact that they may be derived from the Tractrix thus : — Draw a circle with 
radius Q R, centre Q (Fig. 113); estimate latitude on .such circle from R upward; the intercept on 
Q S between consecutive tangents to the Tractrix will be the increment for the arc of latitude 
included between parallels to Q S, drawn through the points of contact of said pair of tangents.* 

On the subject of map construction the student is referred to Craig's Treatise on Projections. 

THE WITCH OF AC4NISI. 

205. If on any line S Q, perpendicular to the diameter of a circle, a point S be so located 
that S Q : A B :: P Q : Q B then S will be a point of the curve called the Witch of Agnisi. Such 
point is evidently on the ordinate P Q prolonged, and vertically below the intersection T of the 
tangent at A by the secant through P. 




The point E, at the same level as the centre 0, is a diameter's distance from the^ latter. 

The tangent at B is an asymptote to the curve. 

The area between the curve and its asymptote is four times that of the circle involved in its 
construction. 

The Witch, also called the Versiera, was devised by Donna Maria Gaetana Agnisi, a brilliant 
Italian lady, who was appointed (1750) by Pope Benedict XIV. to the professorship of mathematics 
and philosophy in the University of Bologna. 

THE CARTESIAN OVAL. 

206. This curve, also called simply a Cartesicvn, after its investigator, Descartes, has its points 
connected with two foci, F' and F", by the relation m p' ± n p" = k c, in which c is the distance 
between the foci while m, n and k are constant factors. 



* Leslie. Geometrical Anal/isis. Edinburgh, 1831. 



72 



THEORETICAL AND PRACTICAL GRAPHICS. 



:rLg. ±±T. 



Salmon states that we owe to Chasles the proof that a thkd focus may be found, sustaming the 

same relation, expressed by an equation of similar form. (See Art. 
209.) 

The Cartesian is symmetrical with respect to the axis — the line 
joining the foci. 

207. To construct the curve from the first equation we may for 




convenience write w p' ± a p' 



h c in tlie form o' ± — o' 

m 



k c 



-; or 



n a' c 

by denoting — by b and — by d it takes the yet more simple form 
p' d= 6 p" = d. Then p" will have two values according as the positive 



d- p' 



or negative sign is taken, being respectively — r 



and 



.' - d 



; the former is for points on the 



inner of the two ovals that constitute a complete Cartesian, while the latter gives points on the 
outer curve. 

d—p' Figr. lis. 



To obtain p" = 



take F' and F" (Fig. 



118) as foci; F' S=d; SK at some random acute 

d 
a,ngle with the axis, and make S H = -■ that 



is, make F' S : S H :: b : 1. Then from F' draw 
an arc </P, of radius less than d, and cut it at 
P by an arc from centre F", radius ST, Tt being 
a parallel to F' H ; then P is a point of the 
inner oval; for St^d — p' and ST=p"; there- 

lore p : d — p : : . : a whence p = — ^— ^. 

208. If an arc x y h he drawn from F' with 
radius, F' x, greater than d, we may find the second 




d 



.iy drawing xQ parallel to F' H to meet HS prolonged; for QS will 



value of p", viz., 

€qual — 7 — , in which p' = F' x. Again using F" as a centre, and a radius Q S =. p", gives points 
R and M of the larger oval. 

The following are the values for the focal radii to the four points where the ovals cut the 
axes. (See Fig. 117.) 

P 



For A, p" = 



«, P = 



d' , / 1 / ni J d -\- h c 

~ = e + p whence p = F A = 



c + p' 



" b, p" = ^^' = c ■ 



p' = F' a = 
p' = F' B = 
p' = F'b = 



1 — 5 


d — 5 c 


1 + 5 


d + b c 


1 + 5 


d — be 



1 — 5 

The construction-arcs for the outer oval must evidently have radii between the values of p' for A 
and B above; and for the inner oval betiveen those of a and b. 



CA R TESIA X VA L S. — CA US TICS. 



IB 




The numerical values from which Fig. 118 was constructed were m = 3 ; ii = 2 ; c=l; />■ = 3. 

209. The Third Focus. Figure 118 illustrates a special case, but in general the method of finding 
a third focus. F'" (not shown), would be to draw a random secant F' r through F' and note the 
points P and G in which it cuts the ovals — these to be taken on the same side of F', as two 
other points of intersection are possible : a circle through P, G and F" would cut the axis in the 
new focus sought. Then denoting by C the distance F' F"', we would find the factors of the original 
equation appearing in a new order, thus, A-p'±np"'= m C, which — for purposes of construction — 
may be written p' ± h' p'" — d' . 

If obtained from the foci /"'and F'" the relation would be m p"' — A'p"=±/iC', in which C 
equals F" F'". Writing this in the form p'" — B p" ^ ± D we have the following interesting cases: 
(a) an ellipse for D positive and B ^ — 1 ; (b) an hyperbola for D positive and 5 = + 1 ; (c) a 
lirnacon for D = C B ; (d) a cardioid for B = + 1 and D = C. 

210. The following method of drawing a Cartesian by continuous motion was ^'-s- ns- 
devised by Prof Hammond: — A string is wound, as shown, around two pulleys 
turning on a common axis ; a pencil at P holds the string taut around smooth 
pegs placed at random at F, and F„ : if the wheels be turned with the same 
angular velocity and the jDencil does not slip on the string it will trace a Cartesian 
having F^ and F., as foci.' 

If the pulleys are equal the Cartesian will become an ellipse ; if both threads 
are wound the same icay around either one of the wheels the resulting curve will be 
an hyperbola. 

211. It is a well-known fact that the incident and reflected ray make equal angles with the 
normal to a reflectiiig surface. If the latter is curved then each reflected ray cuts the one next to 

it. their consecutive intersections giving a curve called a caustic 
by reflection. Probably all have occasionally noticed such a curve 
on the surface of the milk in a glass, when the light was 
properly placed. If the reflecting curve is a circle the caustic is 
the evolutc of a limaron. 

In passing from one medium into another, as from air into 
water, the deflection which a ray of light undergoes is called 
refraction, and for the same media the ratio of the sines of the 
angles of incidence and refi-action {0 and 4>, Fig- 120,) is constant. 
The consecutive intersections of refracted rays give also a caustic, 
which, for a circle, is the evolute of a Cartesian Oval. The proof of this statement' involves the 
property upon which is based the most convenient method of drawing a tangent to the Cartesian, 
viz., that the normal at any point divides the angle between the focal radii into parts whose sines 
are proportional to the factors of those radii in the equation. If, then, we have obtained a point 
G on the outer oval from the relation m p' ± n p" = he we may obtain the tangent at G bj' la3'ing 
off on p' and p" distances proportional to m and n, as Gr find G h, Fig. 118, then bisecting rh 
at / and drawing the normal G j, to which the desired tangent is a jDerpendicular. 

At a point on the inner oval the distance would not be laid off on a focal radius produced, as 
in the case illustrated. 



ig-. 12C- 




1 American Journal of Mathematics, 1S78. 



- Salmon. Hiqhcr Plane Curves. Art. 117. 



74 



THEORETICAL AND PRACTICAL GRAPHICS. 



CASSIAN OVALS. 

212. In the Gassian Ovals or Ovals of Cassini the points are connected with two foci by the 
relation p' p" = I-", i. e., the product of the focal radii is equal to some perfect square. These curves 
have already been alluded to in Art. 114 as, plane sections of the annular torus, taken parallel to 
its axis. 




^ig-- las. 



In Art. 158 one form — the Lemniscate — receives special treatment. For it the constant k'' must 
equal m^, the square of half the distance between the foci. When k is less than m the curve 
becomes two separate ovals. 

213. The general construction depends on the fact that in an}' semicircle the square of an ordinate 
equals the product of the segments into which it divides the diameter. In Fig. 122 take F^ and 
F^ as the foci, erect a peri^endicular F^ S to the axis 
Fi F^ and on it lay off F^ R equal to the constant Jc. 
Bisect F^ F, at and draw a semicircle of radius R. 
This cuts the axis at A and B, the extreme points of 
the curve; for k'' = F^ A X F-^ B. Any other point T 
m.ay be obtained by drawing from F^ a circular arc of 
radius F^ t greater than F^ A ; draw t R, then R x perpen- 
dicular to it; X F^ will then be the p" and F^t the p' for 
four jooints of the curve, which wdll be at the intersection of 
arcs struck from F^ and F.^ as centres and with those radii. 

To get a normal at any point T draw T, then make angle F^ T s = 6 — F^ TO; Ts will be 
the desired line.- 




THE CATENARY. 

214. If a flexible chain, cable or string, of uniform weight per unit of length, be freely 
suspended by its extremities, the curve which it takes under the action of gravity is called a 
Catenary, fi-om catena, a chain. 

A simple and practical method of obtaining a catenary on the drawing-board would be to insert 
two pins in the board, in the desired relative position of the points of suspension, and then attach 
to them a string . of the desired length. By holding the board vertically the string would assume 
the catenary, whose points could then be located with the pencil and joined in the usual manner 
with the irregular curve. Otherwise, if its points are to be located by means of an equation, we 
take axes in the plane of the curve, the i/-axis (Fig. 123) being a vertical line through the lowest 
point T of the catenary, while the x-axis is a horizontal line at the distance m below T. The 
quantity m is called the 'parameter of the curve and is equal to the length of string which represents 
the tension at the lowest point. 



THE CATENARY.— THE LOGARITHMIC SPIRAL. 



75 



The equation of the catenary' is then (/=-—((•"'+ e " 

logarithms" and has the numerical vakie 2.71S"2S18 +. 

By taking successive values of .v equal to m, 2 m, 3 m, 
etc.. we get the following values for y: — 

— (<? + -) which for )?i = unity becomes 1.54308 



) 



in which e is the base of Napierian 

S'i-S- 3.23. 






m... >/ 



.-■ '»■/■. 1 \ 

x=om...y = ^\^e'+ -j 

. "I / X 1 \ 

:c = -Lm...y=^[e^+-j 



3.76217 
10.0676 
27.308 



To construct the curve we therefore draw an arc of 
radius B = m, giving T on the axis of y as the lowest 
point of the curve. 



For :r = B = m 




have y = B P = 1.54308: for x = a = — we have */ = a n= 1.03142. 



The tension at any point P is equal to the weight of a piece of rope of length BP^PC+m. 

At the lowest point the tangent is horizontal. The length of any arc T P is proportional to the 
angle between T C and the tangent P V at the upper extremity of the arc. 

215. If a circle R L B be drawn, of radius equal to m, it may be shown anah^ically that 
tangents P S and Q R, to catenary and circle respectively, from points at the same level, will be 
parallel : also that P S equals the catenary-arc P r T ; S therefore traces the involute of the catenarj^, 
and as SB always equals RO and remains perpendicular to P S (angle R Q being always 90°) 
we have the curve TSK fulfilling the conditions of a tracirix. (See Art. 202.) 

If a parabola, having a focal distance m, roll on a straight line, the focus will trace a catenary 
having m for its parameter. 

The catenary was mistaken Ij}- Galileo for a parabola. In 1669 Jungius jjroved it to be neither 
a parabola nor hj'perbola, but it was not till l(i91 that its exact mathematical nature was known, 
being then established by .James Bernouilli. 



THE LOGARITHMIC OR EQUIAXGUL.iR SPIRAL. 

216. In Fig. 124 we have the curve called the Logarithmic Spiral. Its usual construction is based 
on the jjroperty that any radius vector, as p. which bisects the angle between two other radii, M 
and N, is a mean proportional between them ; i. e., p- = S'' = M X X. 

If 31 and G are points of the spiral we may find an intermediate point K by drawing the 
ordinate K to a semicircle of diameter M + G. A perpendicular through G to G K will then 
give D, another point of the curve, and this construction may be repeated indefinitely. 

Radii making equal angles with each other are evidently in geometrical progression. 

The curve never reaches the pole. 



1 Runkine. Applied Mechanics, Art. 175. 

-In ttie expression 10- = 100 the quantity *'*2" is called the togartthm of 109, it being the exponent of the power to which 
10 must he raised to give 100. Similarlj- 2 would be the logarithm of 61 were 8 the 6ttse or number to be raised to the power 
indicated. 



76 



THEORETICAL AND PRACTICAL GRAPHICS. 



This spiral is often called Equiangular from the fact that the angle is always the same between 
a radius vector and the tangent at its extremity. Upon this property is based its use as the out- 
line for spiral cams and for lobed wheels. 

The name logarithmic spiral is based on the property that 
the angle of revolution is firoportional to the logarithm of the 
radius vector. This is expressed by p = a^, in which 6 is the 
varying angle and a is some arbitrary constant. 

To construct a tangent by calculation divide the hyperbolic 
logarithm ' of the ratio M : K (which are any two radii 
whose values are known) by the angle between these radii, 
expressed in circular measure'; the quotient will be the tangent 
of the constant angle of obliquity of the spiral. 

217. Among the more interesting properties of this curve 
are the following: — 

Its involute is an equal logarithmic spiral. 
Were a light placed at the pole, the caustic — whether by 
reflection or refraction — would , be a logarithmic spiral. 

The discovery of these properties of recurrence led James 
Bernouilli to direct that this spiral be engraved on his tomb, 
with the inscription — Eadem Mutata Resurgo, which, freely trans- 
lated, is — / shall arise the same, though changed. 

Kepler discovered that the orbits of the planets and comets were conic sections having a focus 
at the centre of the sun. Newton proved that they would have described logarithmic spirals as 
they travelled out into space had the attraction of gravitation been inversely as the cube instead of 
the square of the distance. 




THE HYPEBBOLIC OE RECIPROCAL SPIRAL. 

218. In this spiral the length of a radius vector is in inverse ratio to the angle through which 
it turns. 

Like the logarithmic spiral it has an infinite number of convolutions aljout the pole, which it 
never reaches. 

The invention of this curve is attributed to James Bernouilli, who showed that Newton's con- 
clusions as to the logarithmic spiral (see Art. 217) would also hold for the hyperbolic spiral, the 
initial velocity of projection determining which trajectory was -described. s'i.s- iss. 

To oljtain points of the curve divide a circle m 5 8 (Fig. 125) 
into any number of equal parts, and on some initial radius m 
lay off some unit, as an inch; on the second radius 2 take 

7h 01/ 

— ^; on the third — ^, etc. For one-half the angle the radius 

vector would evidently be 2 n, giving a point s outside the 
circle. 

1 



The equation to the curve is 



in which 



is the 




1 To get the hypertolic logarithm of a number multiply Its common logarithm by 2.3026. 
= In circular measure 3G0° = 2 IT r which for r = 1 becomes 6,28318; 180° = 3.14159; 90o = 1.5708; 60 = 
0..'J236 ; 1 ° = 0.0174533. 



. 1.0472 ; 45 o = 0.7854 ; 30 ° : 



TEE HYPERBOLIC SPIRAL.— THE L ITU US. 



77 



radius vector, a some nmiierical constant, and is the angular rotation of /• (in circular measure) 
estimated from some initial line. 

The curve has an asymptote parallel to the initial line and at a distance from it equal to 

- units. 




To construct the spiral fi-om its equation take as the pole (Fig. 126): OQ as the initial line: 

a for convenience, some fi-action, as -: and as our unit some quantitv. say half an inch, that will 

4' 

make - of convenient size. Then taking Q as the initial line make P = - = 2" and draw PR 
a " a 

parallel to OQ for the asymptote. For 6=1. that is. for arc KH = radius OH we have 
?-=-=2". givintj H. — one point of the spiral. Writing the equation in the form r=-.-r and 

expressing various values of 9 in circular measure we get the following: — 

e = .30° = 0.5236; ;■ = J/ = 3'.'8 + : 0= 45° = 0.7854; r=0 -Y = 2 '.'55 ; 
e = m° = 1.5708; r = S = i:-l + : 6 = 180° = 3.14159; r = T= .6366. etc. 
The tangent to the curve at any point makes with the radius vector an angle ^ which is found 

by analysis to sustain to the angle 6 the following trigonometrical relation, tan <t> ^ 6; the circular 

measure of xn&y therefore be found in a table of natural tangents and the corresponding value of 

(^ obtained. 

THE Lixrrs. 



219. The spiral in which the radius vector is inversely proportional to the square root of the 
angle through which it has revolved is called the Lituu-s. This relation is shown bv the equation 

r = 7^1 also written a.- 6 = - . 
ay 6 r 

When = we find r = oo . making the initial line an asymptote to the curve. 

In Fig. 127 take Q as the initial line. as the pole, a = 2, and our unit a three-inch line : 

then - = 1;". 
a 



THEORETICAL AND PRACTICAL GRAPHICS. 



For e = 90° = TT (in circular measiu-e 1.5708) we have r= M=l'!-2 +. - For 6=1 we have 
the radius T making an angle of 57 ? 29 + with the initial line, and in length equal to - units, 



i. e., U". For 6=45°= -f (or 0.7854) r will be OR=V:i +. Then 0H = 



R 



for in 



rotating to H the radius vector passes over four 45° angles, and the radius must therefore be one- 
half what it was for the first 45° described. Similarlj^ K = —^; OM — —^, etc.; this relation 
enabling the student to locate an_y number of points. 




To draw a tangent to the curve Ave emjoloy the relation tan ^=26, <^ being the angle made 
bj' the tangent line with the radius vector, while is the angular rotation of the latter, in circular 
measure. 



BRUSH TINTING AND SHADING. 79 



CHAPTER VI. 



TINTIJJG — FLAT AND GRADUATED. — MASONEY, TILING, "WOOD GRAINING, RIVER-BEDS AND 
OTHER SECTIONS, WITH BRUSH ALONE OR IN COMBINED BRUSH AND LINE WORK. 

220. Brush-work, with ink or colors, is either flat or graduated. The former gives the effect of 
a flat surface parallel to the paper on which the drawing is made, while graded tints either show 
cun-ature, or — if indicating flat surfaces — represent them as inclined to the paper, i.e., to the plane 
of projection. For either, the paper should be, as pre^dously stated (Arts. 41 and 44) cold-pressed and 
stretched. 

The surface to be tinted should not be abraded by sponge, knife or rubber. 

221. The liquid emploj-ed for tinting must be tree from sediment; or at least the latter, if 
present, must be allowed to settle, and the brush dii3ped only in the clear portion at the top. Tints 
may, therefore, best be mixed in an artist's water-glass, rather than in anything shallower. In case 
of several colors mixed together, however, it would be necessary to thoroughly stir ujj the tint each 
time before taking a brushful. 

A tint 23repared trom a cake of high-grade India ink is far superior to any that can be made 
by using the ready-made liquid drawing inks. 

222. The size of brush should bear some relation to that of the surface to be tinted; large 
brushes for large surfaces and vice versa. The customary error of beginners is to use too small and 
too dr}' a brush for tinting, and the reverse for shading. 

223. Harsh outlines are to be avoided in brush work, especially in handsomely shaded drawings, 
in which, if sharply defined, they would detract from the general effect. This will become evident 
on comparing the spheres in Figs. 1 and 4 of Plate II. 

Since tinting and shading can be successfully done, after a little practice, with onlj' pencilled 
limits, there is but little excuse for inking the boundaries; but if, for the sake of definiteness, the 
outlines are 'inked at all it should be before the tinting, and in the finest of lines, preferably of 
"water -proof" ink; although any ink will do provided a soft sjDonge and plenty of clean water be 
applied to remove any excess that will "run." The sponge is also to be the main reliance of the 
draughtsman for the correction of errors in brush work; the water, however, and not the friction to 
be the active agent. An entire tint may be removed in this way in case it seems desirable. 

224. AMien beginning work incline the board at a small angle, so that the tint will flow down 
after the brush. For a flat, that is, a unijorra tint, start at the upper outline of the surface to be 
covered, and with a brush fuU, yet not surcharged — which would prevent its coming to a good point — 
pass lightly along from left to right, and on the return carry the tint down a little farther, making 
short, quick strokes, with the brush held almost perpendicular to the paper. Advance the tint as 
evenl}' as possible along a horizontal line; work quickly betireen outlines, but more slowly along 
outlines, as one should never overrun the latter and then resort to "trimming" to conceal lack of 
skill. It is possible for any one, with care and practice, to tint to yet not over boundaries. 

The advancing edge of the tint must not be allowed to dry until the lower boundary is reached. 



80 



THEORETICAL AND PRACTICAL GRAPHICS. 



No portion of the paper, however small, should he missed as the tint advances, as the work is 
likely to be spoiled by retouching. 

Should any excess of tint be found along the lower edge of the figure it should be absorbed 
by the brush, after first removing the latter's surplus by means of blotting paper. 

To get a dark effect several medium tints laid on in succession, each one drying before the 
next is applied, give better results than one dark one. 

The heightened effect described in Art. 72, viz., a line of light on the ui:)per and left-hand edges, 
may be obtained either (a) by ruhng a broad line of tint with the drawing-pen at the desired 
distance from the outline, and instantly, before it dries, tinting from it with the brush; or (b) by 
ruling the line with the pen and thick Chinese White. 

225. A tint will spread much more evenly on a large surface if the paper be first slightly 
dampened with clean water. As the tint will follow the water, the latter should be limited exactly 
to the intended outlines of the final tint. 




226. Of the colors frequently used by engineers and architects those which work best for flat 
efiects are carmine, Prussian blue, burnt sienna and Payne's gray. Sepia and Gamboge, are, fortunately, 
rarely required for uniform tints; but the former works ideally for shading bj^ the "dry" process 
described in the next article; and its rich brown gives effects unapproachable with anything else. 
It has, however, this peculiarity, that repeated touches ujDon a spot to make it darker produce the 
opposite effect, unless enough time elapses between the strokes to allow each addition to dry thoroughly. 

227. For elementarj' practice with the brush the student should lay flat washes, in India tints 
on from six to ten rectangles, of sizes between 2" x 6" and 6" X 10". If successful with these 
his next work may be the reproduction of Fig. 128, in which H, V, P and S denote horizontal, 
vertical, profile and section planes respectively. The figure should be considerably enlarged. 

The plane V may have two washes of India ink; H one of Prussian blue; P one of burnt 
sienna, and S one of carmine. 

The edges of the planes H, V and P are either vertical or inclined 30° to the horizontal. 



BRUSH TINTING AND SHADING. 



81 



For the section -plane assume a and 7)!, at pleasure, gi^'ing direction nm, to which JR and TX 
are parallel. A horizontal, mz, through m gives z. From n a horizontal, ny, gives y on ab. 
Joining y with 2 gives the "trace" of S on V. 

228. Figures 129 and 130 illustrate the use of the brush in the representation of masonry. 
The former ma}' be altogether in ink tints, or in medium burnt umber for the front rectangle of 



-^ '■ "~ ■■■■ " '-■■■■■-" - - ■ 


- — — — ^T — T^ — " — ""^^ "' '~""'?r* ■ 


:^' "" 


j 


'. ' ' 







each stone, and dark tint of the same, directly from the cake, for the bevel. Lightly pencilled 
limits of bevel and rectangle will be needed; no inked outlines required or desirable. 

The last remark applies also to Fig. 130, in which " quarry - faced " ashlar niasomy is represented. 
If properly done, in either burnt umber or sepia, this gives a result of great beauty, especially 
effective on the piers of a large drawing of a bridge. 

The darker portions are tinted directly from the cake, and are purposely made irregular and 
"jagged" to reproduce as closely as possible the fractured appearance of the stone. 




Two brushes are required when an ''over -hang" or jutting portion is to be represented, one \vith 
a medium tint, the other with the thick color, as before. An irregular line being made with the 
latter, the tint is then softened out on the lower side with the point of the brush having the lighter 
tint. A light wash of the intended tone of the whole mass is quickly laid over each stone, either 
before or after the irregularities are represented, according as an exceedingly angular or a somewhat 
softened and rounded effect is desired. 



82 



THEORETICAL AND PRACTICAL GRAPHICS. 



229. Designs in tiling are excellent exercises, not only for brush work in flat tints, but also — in 
their preliminary construction — in precision of line work. The suj^erbly illustrated catalogues of the 
Minton Tile Works are, unfortunately, not accessible by all students, illustrating as they do, the finest 
and most varied work in this line, both of designer and chromo-lithographer; but it is quite within 
the bounds of possibility for the careful draughtsman to closely aiaproach if not equal the standard 
and general appearance of their work, and as suggestions therefor Figs. 131 and 132 are presented. 

230. In Fig. 131 the upper boundary, adhh, of a rectangle is divided at o, 6, c, etc., into 
equal spaces, and through each point of division two lines are drawn with the 30° triangle, as 6 a; 
and hr through h. The oblique lines terminate on the sides and lower line of the rectangle. If 
the work is accurate — and it is worthless if not — any vertical line as mn, drawn tlirough the inter- 
section, m, of a pair of oblique lines, will pass through the intersection of a series of such pairs. 

^'i-S- 131- 




The figure shows three of the possible designs whose construction is based on the dotted lines 
of the figure. For that at the top and right, in ^^'hich horizontal rows of rhombi are left white, we 
draw -^-ertical lines as s q and m n from the lower vertex of each intended white rhombus, continuing 
it over two rhombi, when another white one will be reached. The dark faces of the design are to 
be finally in solid black, previous to which the lighter faces should be tinted with some drab or 
brown tint. The pencilled construction lines would necessarily be erased before the tint Avas laid on. 

The most opaque effect in colors is obtained by mixing a large portion of Chinese white 
with the water color, making what is called by artists a "body color." Such a mixture gives a 
result in marked contrast with the transparent effect of the usual wash; but the amount of white 
used should be sufficient to make the tint in reality a paste, and no more should be taken on the 
brush at one time than is needed to cover one figure. 

Sepia and Chinese white, mixed in the projDcr proportions, give a tint which contrasts most 
agreeably with the black and white of the remainder of the figure. The star design and the hexagons 
in the lower right-hand corner result from extensions or modifications of the construction just 
described which will become evident on careful inspection. 



TINTING. — BRUSH SHADING. 



83 



231. Fig. 132 is a Minton design with which many are famihar, and which affords opportunity 
for considerable variety in finish. Its construction is almost self-evident. The equal spaces, ab, cd, 
mn — which may be any width, x, — alternate with other equal spaces be, which may preferably be 
about 3x in width. Lines at 45°, as indicated, complete the preliminaries to tinting. 

E'igr- 132. 




The octagons may be in Prussian blue, the hexagons in carmine, and the remainder in white 
and black, as shown; or browns and drabs may be employed for more subdued effects. 



SHADING. 



232. For shading, by graduated tints, provide a glass of clear water in addition to the tint; 
also an ample supjily of blotting jiaper. 

The water -color or ink tint may be considerably darker than for flat tinting; in fact, the darker 
it is, provided it is clear, the more rapidly can the desired effect be obtained. 

The brush must contain much less liquid than for flat work. 

Lay a narrow band of tint quickly along the part that is to be the darkest, then dip the brush 
into clear water and immediate^ aj^ply it to the blottei', both to bring it to a good point and to 
remove the surjjlus tint. With the now once -diluted tint carrj' the advancing edge of the band 
slightly farther. Repeat the oijeration until the tint is no longer discernible as such. 

The process may be ■ repeated from the same starting point as man}' times as necessary to 
produce the desired effect; but the work should be allowed to drj'' each time before laying on a 
new tint. 

Any irregularities or streaks can easih' be removed after the work dries, by retouching or 
"stippling" with the point of- a fine brush that contains but little tint — scarcely more than enough 
to enable the brush to retain its i^oint. For small work, as the shading of rivets, rods, etc., the 
process just mentioned, which is also called "dry shading." is especially adapted, and, although 
somewhat tedious, gives the handsomest effects i^ossible to the draughtsman. 

233. Where a good, general effect is wanted, to be obtained in less time than would be required 
for the preceding processes, the method of over-lapping flat tints may be adopted. A narrower band 
of dark tint is first laid over the part to be the darkest. A\'hen dry this is overlaid by a liroader 
band of lighter tint. A yet lighter wash follows, Ijeginning on the dark portion and extending still 
farther than its predecessor. The process is reijeated with further diluted tints until the desired 
effect is obtained. 

Faintly -pencilled lines may l)e drawn at the outset as limits for the edges of the tints. 



84 



THEORETICAL AND PRACTICAL GRAPHICS. 



This method is better adapted for large work, that is not to be closely scrutinized, than for 
drawings that deserve a high degree of finish. 

234. As to the relative position and gradation of the lights and shades on a figure, the student 
is referred to Arts. 78 and 79 and the chapter on shadows; also to the figures of Plate II, which 
may serve as examples to be imitated while the learner is acquiring facility in the use of the brush, 
and before entering ujjon constructive work in shades and shadows. Fig. 3 of Plate II may be 
undertaken first, and the contrast made yet greater between the upper and lower boundaries. Fig. 1 
(Plate II) requires no explanation. In Fig. 133 we have a wood -cut of a sphere, with the theo- 
retical dark or "shade'' line more sharply defined than in the spheres on the plate. 



^ig-- 133- 



^ig-. a.3-4. 






J 




'A drawing of the end of a highly -ijolished revolving shaft, or even of an ordinary metallic disc, 
would be shaded as in Fig. 184. 

Fig. 2 (Plate II) represents the triangular -threaded screw, its oblique surfaces being, in mathe- 
matical language, warped helicoids, generated by a moving straight line, one end of which travels along 
the axis of a cylinder while the other end traces or follows a helix on the cylinder. 

The construction of the helix having already been given (Art. 120) the outlines can readily be 
drawn. The method of exactly locating the shadow and shade lines will be found in the chapter 
on shadows. 

Fig. 4 (Plate II), when compared with Fig. 91, illustrates the possibilities as to the 
representation of interesting mathematical relations. The fact may again be mentioned, on the 
princii:)le of "line upon line," as also for the benefit of any who may not have read all that has 
preceded, that the spheres in the cone are tangent to the oblique plane at the foci of the elliptical 
section. The peculiar dotted effect in this figure is due to the fact that the original drawing, of 
which this is a photographic reproduction by the gelatine process, was made with a lithographic 
craj'on upon a special pebbled paper much used by lithographers. The original of Fig. 1, on the 
other hand, was a brush -.shaded sphere on Whatman's jDaper. 

235. Fig. 5 (Plate II) shows a "Phoenix column," the strongest form of iron for a given weight, 

for sustaining compression. The student is familiar with it as an 
element of outdoor construction in bridges, elevated railroads, etc.; 
also in indoor work in many of the higher office buildings of our 
great cities. 

By drawing first an end view of a Phoenix column, similar to 
that of Fig. 135, we can readily derive an oblique view like that of 
the jjlate, by including it between parallels from all points of the 
former. The ijroportions of the columns are obtainable from the 
tables of the company. 

Fig. 135 is a cross- section of the 8 -segment column, the shaded 
portion showing the minimum and the other lines the maximum 
size for the same inside diameter. 



^ig-. 13E. 




MATERIALS OF CONSTRUCTION. 



85 



In a later chapter the proportions of other forms of structural iron will be found. Short 



lengths of any of these, if shown in oblique view, are good subjects for the 
brush, especially for "dry" shading, the effect to be aimed at being that 
of the rail section of Fig. 136. 

236. When some f)articular material is to be indicated, a flat tint of the 
proper technical color (see Art. 73) should be laid on with the brush, either 
before or after shading. When the latter is clone with sepia it is jjrobably 
safer to lay on the flat tint first. 

A darker tint of the technical color should always be given to a cross- 
section. For blue -printing, a cross -section may be indicated in solid black. 



S^S. i3S. 




WOOD. — RIVER - BEDS. — M.iSOXRY, ETC. 

237. While the engineering draughtsman is ordinarih' so pressed for time as not to be able to 
give his work the highest finish, yet he ought to be able, when occasion demands, to obtain 
both natural and artistic effects; and to conduce to that end the writer has taken pains to illustrate 
a number of ways of representing the materials of construction. .\1 though nearly all of them may 
be — and in the cuts are — represented in black and white (with the exception of the wood - graining 
on Plate II), j-et colors, in combined brush and line work, are preferable. The student will, however, 
need considerable practice with pen and ink before it will be worth while to work on a tinted figure. 

238. Ordinarily, in representing wood, the mere fact that it is wood is all that is intended to 
be indicated. This may be done most simply by a series of irregular, approximately - i^arallel lines, 
as in Fig. 10 or as on the rule in Fig. 17, page 12. Make no attempt, however, to have the grain. 
very irregular. The natural unsteadiness of the hand, in drawing a long line toward one continu- 
ously, will cause almost all the irregularity desired. 

If a better effect is wanted, yet without color, the lines vaay be as in Fig. 107, which represents 
hard wood. 

In graining, the draughtsman should make his lines toward himself, standing, so to S23eak, at the 
end of the plank upon which he is working. 

The splintered end of a plank should be sharjjly toothed, in contradistinction to a metal or 
stone fracture, which is what might be called smoothly irregular. 

239. An examination of anj' piece of wood on which the grain is at all marked will show 
that it is darker at the inner vertex of any marking than at the outer point. Although this 
difference is more readih' produced with the brush, yet it may be shown in a satisfactorj' degree 
with the pen, by a series of after -touches. 

240. If we fiU the pen with a rather dark tint of the conventional color, draw the grain as in 
the figures just referred to, and then overlay all with a medium flat wash of some properly chosen 
color, we get effects similar to those of Plate II. 

On large timber- work the preliminary graining, as also the final wash, may be done altogether 
with the brush; as was the original of Fig. 9, Plate II. 

End -^-iews of timbers and planks are conventionally represented by a series of concentric free- 
hand rings in which the spacing increases with the distance from the heart; these are overlaid with 
a few radial strokes of darker tint. In ink alone the appearance is shown in Figs. 39 and 115. 

241. The color -mixtures recommended b\- different writers on wood graining are something short 
of infinite in number; but with the addition of one or two colors to those listed in the draughts- 
man's outfit (Art. 56 j one should be able to imitate nature's tints very closely. 



86 



THEORETICAL AND P RA C T I CA L G RAP HIC S. 





COURSED RUBBLE MASONRY 
Light India inli. 



No hard-and-fast rule as to the proportions of the colors can be given. In this connection we 
may quote Sir Joshua Re3'nolds' reply to the one who inquired how he mixed his paints. "With 
brains," said he. One general rule, however; always employ delicate rather than glaring tints. 

Merely to indicate wood with a color and no graining use burnt sienna, the tint of Figs 7, 8 
and 10 of Plate II. 

Drawing from the writer's experience and from the suggestions of various experimenters in this 
line the following hints are presented: — 

In every case grain first, then overlay with the ground tint, which should always be much lighter 
than the color used for the grain. If possible have at hand a good specimen of the wood to be 
imitated. 

Hard Pine: Grain — burnt umber with either carmine or crimson lake; for overlay add a little 
gamboge to the grain -tint diluted. 

Soft Pine: Gamboge or yellow ochre with a small amount of burnt sienna. 

Black Walnut: Grain — burnt umber and a very little dragon's blood; final overlay of modified 
tint of the same or with the addition of Payne's gray. ^^s- -ysv. 

Oak: Grain — burnt sienna; for overlay, the same, with 
yellow ochre. 

Chestnut: Grain — burnt umber and dragon's blood; over- 
lay of the same, diluted, and with a large proportion of gam- 
boge or light yellow added. 

Spruce: Grain — burnt umber, medium; add yellow ochre 
for the overlay. 

Mahogany: Grain — -burnt sienna or umber with a small 
amount of dragon's blood; dilute, and add light yellow for 
the overlay. 

Rosewood: Grain — replace the dragon's blood of mahogany- 
grain by carmine, and for overlay dilute and add a little 
Prussian blue. 

242. River-beds in black and white or in colors have 
been already treated in Art. 26, to which it is only neces- 
sarj' to add that such sections are usually made quite narrow, 
and, preferably — if in color — shaded quite abruptly on the 
side ojjposite the water. 

243. The sections of masonry, concrete, brick, glass and vul- 
canite, given on page 25 as pen and ink exercises, are again 

presented in Fig. 137, for reproduction in combined brush and line 
work. The appropriate color is indicated under each section. 

244. Masonry constructions may be broadly divided into rubble 
and ashlar. 

In ashlar masonry the bed -surfaces and the joints (edges) are 
shaped and dressed with great care, so that the stones ma)' not 
only be placed in regular layers or courses, but often fill exactly 
some predetermined place, as in arch construction, in which case the determination of their forms 
and the derivation of the patterns for the stone-cutter involves the application of the Descriptive 
Geometry of Monge. (Art. 283). 




RUBBLE MASONRY 
Liglit India Ink. 



VULCANITE 
India Ink, 





BRICK 
Venetian Red. 



CONCRETE 
Yellow Ochre. 



FLs- J.3S- 




iiMii'b,M<ii[[M[[|| .MJ,i w.'|HMW;Mi„ H.U..Mil .itiAII, 



REP RE SEN TA TI ON OF MA S NR Y. 



87 



^'igr. i-ao. 



Rubble work, however, consists of constructions involving stones mainly "in the rough,"' but may 
be either coursed or uucoursed. Fig. 138 is a neat example of uncoursed though partially dressed 
or "hammered" rubble. In .section, as shown in Fig. 137, it is merely necessary to rule section- 
lines over the boundaries of the stones — a remark apjjlying equally to ashlar masonry. 
^^' ^ ' The other examples in this chapter 

are of a.shlar, mainly "quarry -faced," 

that is, with the front nearlj- as rough 

as when quarried. A beveled cir 

"chamfered" ashlar is .shown in Figs. 

129 and 140, the latter .shaded in what 

is i)robably the most effective way for 

smaU work, viz., with dots, the effect 

depending upon the number, not the 

size of the latter. 

Only a careful examination of the 
kind and position of the lines in the other figures on this j^age will disclose the secret of the variety 
in the effects produced. For the handsomest results with any of these figures the pen-work — 




p 




ImM 


m 


(PVfi 


M 






iiBBsh 


^^ 


i .;,■, ',v, :^M 



^iir- 3--41- 




E^ig-. 1-3:2. 







ifc^...r)^::^,...^A 



whether dotting or "cross-hatching" — should be preceded by an undertone of either India ink, 
umber, Payne's gray, cobalt or Prussian blue, according to the kind of stone to be represented. 



^'ig-- l^j. 




^Lg-- i-a-S:. 




For slate use a pale blue; for brown free-stone either an umber or sepia; while for stone in 
general, kind immaterial, use India ink. 



88 THEORETICAL AND PRACTICAL GRAPHICS. 



C HAPTBB VII. 

-FEEE-HAND AND MECHANICAL LETTERING. — PROPORTIONING OF TITLES. 

245. Practice in lettering forms an essential part of the elementary work of a draughtsman. 
Every drawing has to have its title, and the general effect of the result as a whole depends largely 
upon the quality of the lettering. 

Other things being equal, the expert and rapid draughtsman in this line has a great advantage 
over one who can do it but slowl3^ For this reason free-hand lettering is at a high premium, and 
the beginner should, therefore, aim not only to have his letters correctly formed and properly spaced, 
but, as far as possible, to do without mechanical aids in their construction. When under great 
pressure as to time it is, however, perfectly legitimate to emjDloy some of the mechanical expedients 
used in large establishments as "short cuts" and labor- savers. Among these the jsrincipal are 
"tracing" and the use of rubber tyjjes. 

246. To trace a title one must have at hand complete printed alphabets of the size of type required. 
Placing a piece of tracing-paper over the letter wanted, it is traced with a hard pencil, the paper 
then slipped along to the next letter needed, and the process repeated until the words desired have 
been outlined. The title is then transferred to the drawing by first running over the lines on the 
back of the tracing-paper with a soft jjencil, after which it is only necessary to re-trace the letters 
with a hard pencil, on the face of the transfer-paper, to find their outlines faintly yet sufficiently 
indicated on the paj^er underneath. Carbon pajjer may also be used for transferring. 

247. The process just described would be of little service to a ready free-hand draughtsman, 
but with the use of rubber types, for the wxjrds most frequently recurring in the titles, a merely 
average worker may easily get results which — in point of time — cannot be exceeded by any other 
method. When employing such types either of the following ways may be adopted : (a) a light 
impression may be made with the aniline ink ordinarily used on the pads, and the outlines then 
followed and the "filling in" done either with a writing -pen* or fine-j^ointed sable -hair brush; or 
(b) the impression may be made after moistening the types on a pad that has been thoroughly wet 
with a light tint of India ink. The drawing -ink must then be immediately applied, free-hand, 
with a Falcon pen or sable brush, before the type -impression can dry. The pen need only be passed 
down the middle of a line, as on the dampened surface the ink will spread instantly to the outlines. 

248. The educated draughtsman should, however, be able not only to draw a legible title of the 
simple character required for shop -work, and in which the foregoing expedients would be mainly 
serviceable, but be iDrejiared also for work out of the ordinary line, and, if need be, quite elaborate, 
as on a competitive drawing. Such knowledge can only be gained by careful observation of the 
forms of letters, and considerable practice in their construction. 

No rigid rules can be laid down as to choice of alphabets for the various jDossible cases. 

Common- sense, custom and a natural regard for the "fitness of things" are the determining factors. 

Obviously rustic letters would be out of place on a geometrical drawing, and other incongruities 



*Ket'er to Art. 27 with regard to the pens to be used for the various styles of letters. 



DESIGNING OF TITLES. 89 

will naturallj' suggest themselves. In addition to the hints in Art. 27 a few general principles and 
methods may, however, be stated to the advantage of the beginner, who should also refer to the 
special instructions given in connection with certain specimen alphabets at the end of this work. 

249. In the first place, a title should be symmetrical with respect to a vertical centre-line, a 
rule which should be violated but rarely, and then, usually, when the title is to be somewhat fancy 
in design, as for a magazine cover. 

lementap|j Flates 

drawn by QTorflanif ©an (Korbar ^+ +hE 

LEAniNB TECHNICAL SCHQCL 

Jan. — June, 3001. 

250. If it be a complete as distinguished from a "partial or S(t6- title it will answer the following 
questions which would naturally arise in the mind of the examiner: — 

What is it? — Where done? — By whom? — When? — On what scale? 
In answering these questions the relative valuation and importance of the lines are expressed by 
the sizes and kinds of type chosen. This is a j^oint requiring most careful consideration, as the final 
effect depends largel_y upon a proper balancing of values. 

— ^-^ OF -«-^— 

PERFECTION SUSPENSION BRIDGE 

G-DDdwin^ MackenziE V Cartwright 

•^-^-s— ?s MINNEAPOLIS. MINN, s^— i-^<— 



Scale 4 Ft. = I IN. JVin© 1©? 2©00. Jose Mahtinez. Del. 

251. The "By whom?" may cover two possibilities. In the case of a set of drawings made in 
a scientific school it would refer to the draughtsman, and his name might projjerly have considerably 
greater prominence than in any other case. The upper title on this page is illustrative of this point, 
as also of a symmetrical and balanced arrangement, although cramped as to space, vertically. 

Ordinarily the " By whom ? " will refer to the designer, and the draughtsman's name ought to 
be comparatively inconspicuous, while the name of the designer should be given a fair degree of 
prominence. This, and other important points to be mentioned, are illustrated in the preceding 



90 THEORETICAL AND PRACTICAL GRAPHICS. 

arrangement, printed, like the upper title, from types of which comiDlete alphabets will be found at 
the end of this work. 

252. The abbreviation Del., often placed after the draughtsman's name, is for Delineavit — He drew 
it — and does not indicate what the visitor at the exhibition suj^i^osed, that all good draughtsmen 
hail from Delaware. 

253. The best designed titles are either in the form of two truncated pyramids having, if j)OS- 
sible, the most important line as their common base, or else ellii^tical in shape. 

254. The use of capitals throughout a line dej)ends upon the style of t3'pe. . It gives a most 
unsatisfactory result if the letters are of irregular outline, as is amply evidenced by the words 

each letter of which is exquisite in form, but the combination almost illegible. Contrast them with 
the same style, but in capitals and small letters: — 



Msrfjaniral J^raming. 



■//Mt 



255. As to spacing, the visible white spaces between the letters should be as nearly the same 
as possible. In this feature, as in others, the draughtsman can get much more jjleasing results than 
the printer, since the latter usually has each letter on a separate piece of metal, and can not adjust 
his space to any particular combination of letters, such as FA, L V, W A or A V, where a better 
effect would be obtained by jDlacing the lower part of one letter under Fig-. ±^e. 

the upper part of the next. This is illustrated in Fig. 146, which may ~ V* V 
be contrasted with the printer's best spacing of the separate types for ' 

the A and W in the word " Drawings " of the last title. 

256. The amount of space between letters will depend upon the length of line that the word or 
words must make. If an important word has few letters they should be " spaced out," and the 
letters themselves of the " extended " kind, i. e., broader than their height. The following word will 
illustrate. The characteristic feature of this type, viz., heavy horizontals and light verticals, is com- 
mon to all the variations of a fundamental form frequently called Italian Print. 

^ I^ I HD <3- E . 

When, on the other hand, many letters must be crowded into a small space, a "condensed" 
style of letter must be adopted, of which the following is an example: 

Pennsylvania Railroad. 

257. While the varieties of letters are very numerous 3'et they are all but changes rung on a 
few fundamental or basal forms, the most elementary of which is the 

GOTHIC. ALSO CALLED HALF- BLOCK. 

Letters like B, O, etc., which have, usually, either few straight parts or none at all, may, for 
the sake of variety as also for convenience of construction, be made partially or wholly angular; in 
the latter case the form is called Geometric Gothic by some tyjDe manufacturers. It is only appropriate 
for work exclusively mechanical. The rounded forms are preferable for free-hand lettering. 



LETTERING. 



91 



The following complete Gothic alphabet is so constructed that whether designed in its "con- 
densed" or "extended" form the proper proportions may be easily preserved. 




C < ■■! V i. 



Taking all the solid parts of the letters at the same width as the I, we will find any letter of 
average width, as U, to be twice that unit, plus the opening between the uprights, which last, being 
indeterminate, we may call x, making it small for a "condensed" letter, and broad as need be for 
an "extended" form. 

The word march would foot up 5 U + 3, disregarding— as we would invariably— the amount the 
foot of the R projects beyond the main right-hand outline of the letter. In terms of x this makes 
.5x-|-13, as U = a;-)-2. Allowing spaces of 1^ unit width between letters adds 5 to the above, making 
5X-I-18 for the total length in terms of the I. Assuming x equal to twice the unit we would 
have the whole word equal to twenty -eight units; and if it were to extend seven inches the width 
of the solid parts would therefore be one -quarter of an inch. 

Where the width of a letter is not indicated it is assumed to be that of the U. The W is 
equal to 2U — 1. This relation, however, does not hold good in all alphabets. 

The angular corners are drawn usually with the 45° triangle. 

The guide -Hues show what points of the various letters are to be found on the same level, and 
should be but faintly pencilled. 

As remarked in Art. 27, the extended form of Gothic is one of the best for dimensioning and 
lettering ivorking drawings, and is rapidly coming into use by the profession. 

258. The Full -Block letter next illustrated is easier to work with than the Gothic in the matter of 
preliminary estimate, as the width of each letter— in terms of unit squares— is evident at a glance. 

The same word march would foot up twenty -seven squares without allowing for spaces between 
letters. Calling the latter each two we would have thirty -five squares for the same length as before 
(seven inches), making one -fifth of an inch for the width of the solid parts. For convenience the 
widths of the various letters are summarized: 

1 = 3; C,G,0,Q,S,Z = 4; A, B, D,E,F, J,L, P, R,T,& = 5; H,K,N,U, V,X, Y = 6; M = 7; W = 8. 

259. In case the preliminary figuring were only approximate and there were but two words in 
the line, as, for example. Mechanical Drawing, a safe method of working would be to make a fair 
allowance for the space between the words, begin the first word at the calculated distance to the 
left of the vertical centre-line, complete it, then work the second word backward, beginning with the; 



92 



THEORETICAL AND PRACTICAL GRAPHICS. 



G as far to the right of the reference line as the M was to the left. On completing the second 
word any difference between the actual and the estimated length of the words, due to over- or 
under -width of such letters as M, W and I, will be merged into the space between the words. 

U.crn.j.Lj.ETn 





■H|gW-..,.,... 






•iB:i;}i£? 





■Fxs- ISO- 



With three words in a line the same method might be adopted, the middle word being easily 
placed half way between the otliers, which, by this method of construction would not only begin 
correctly but also terminate where they sliould. 

260. Note particularly that the top of a B is always slightly smaller than the bottom; E-ig-. i-is. 
similarly with the S. This is made necessary by the fact that the eye seems to exaggerate /->j ^-^ 
the upper half of a letter. To get an idea of the amount of difference allowable compare ^ ^ 
the following equal letters printed from Roman tj'pe, condensed. Although not so important 

in the E, some difference between top and bottom may still to advantage be made. Another refine- 
ment is the location of the horizontal cross-bar of an A slightly below the middle of the letter. 

261. While vertical letters are most frequently iised, yet no handsomer effect can be ol;)tained 
than by a well - executed inclined letter. The angle of inclination should be about 70°. 

Beginners usually fail sadly in their first attempt 
with the A and V, one of whose sides they give the 
same slant as the upright of the other letters. In point 
of fact, however, it is the imaginary (though, in the 
construction, pencilled) centre-line which should have that 
inclination. See Fig. 150. 

In these forms — the Roman and Italic Roman — the union of the light horizontals or "seriffs" 

with the other parts is in general effected by means of fine arcs, called "fillets," drawn free-hand. 

On many letters of this alphal^et some lines will, however, meet at an angle, and only a careful 

examination of good models will enable one to construct correct forms. Upon the size of the fillets 

the appearance of the letter mainly depends, as will be seen by a glance at Fig. 151, which repro- 

■s^i-s- 151. duces, exactly, the N of each of two leading alphabet books. If the fillets 

^ y "m Y' round out to the end of the spur of the letter, a coarse and bulky appear- 

/ %/ / %/ ance is evidently the result; while a fine curve, leaving the straight 

horizontals projecting beyond them, gives the finish desired. This is further 

illustrated by No. 23 of the alphabets appended, a type which for clearness and elegance is a triumph 

of the founder's art. As usually constructed, however, the D and R are finished at the top like the P. 




94 



THEORETICAL AND PRACTICAL GRAPHICS. 



■E-i-S- iSS. 



262. The Roman alphabet and its incHned or italic form are much used in topographical work. 
A text -book devoted entirely to the Roman alphabet is in the market, and in some works on 

topographical drawing very elaborate tables of proportions for the letters are presented; these answer 
admirably for the construction of a standard alphabet, but in practice the proportions of the model 
would be preserved by the draughtsman no more closely than his eye could secure. Usually the 
small letters should be about three- fifths the height of the capitals. Except when more than one- 
third of an inch in height, these letters should be entirely free-hand. 

263. When a line of a title is curved no change is made in the forms of the letters; but if of 
a vertical, as distinguished from a slanting or italic type, the centre-line of each letter should, if 
produced, pass through the centre of the curve. 

Italic letters, when arranged on a curve, should have their centre-lines inclined at the same angle 
to the normal (or radius) of the curve as they ordinarily make with the vertical. 

264. An alphabet which gives a most satisfactory appearance, yet can be constructed with great 
rapidity, is what we may call the "Railroad" type, since the public has become familiar with it 
mainly from its frequent use in railroad advertisements. 

The fundamental forms of the small letters, with the essential construction lines, are given in 
rectangular outline in the complete alphabet on the preceding page, with various modifications thereof 
in the words below them, showing a large number of possible effects. 

At least one plain and fancy capital of each letter is also to be found on the same page, with in 
some instances a still larger range of choice. 

No handsomer effects are obtainable than with this alphabet, when brush tints are employed for 
the undertone and shadows. 

265. For rapid lettering on tracing-clotli, Bristol 
board or any smooth - surfaced paper a style long used 
abroad and increasing in favor in this country is that 
known as Round Writing, illustrated by Fig. 152, and 
for which a special text -book and pens have been prepared by F. Soennecken. The pens are 

stubs of various widths, cut off obliquely, and when in 
use should not, as ordinarily, be dipped into the ink, but 
the latter should be inserted, by means of another pen, 
between the top of the Soennecken pen and the brass 
"feeder" that is usually slipped over it to regulate the flow. 
The Soennecken Round Writing Pens are also by far 
the best for lettering in Old English, German Text and 
kindred types. 
The improvement due to the addition of a few straight lines to an ordinary title will become 

evident by comparing Figs. 

153 and 154. The judicioxis ~ 

use of "word ornaments," 

such as those of alphabets 

33, 42, 49, and of several of 

the other forms illustrated, 

will greatly enhance the 

appearance of a title with- 
out materially increasing the time expended on it. This is illustrated in the lower title on page 89. 



QJloiiMd Wu^inq 



E-ig-. 153- 



plernen'ary pare^ 



^ eeh 



^ 



anieal 



D 



ra^vind 



E'igr- 3.5-4. 



D rawl n ^- 



V 


ee_ 


1 


■ an lea 







DESIGNS FOR BORDERS. 



95 




96 



THEORETICAL AND PRACTICAL GRAPHICS. 



V 



~:l 



51 4i3 



ffi 



/ 



u 



266. Bordeis. Another effective adjunct to a map or other drawing is a neat border. It should 
be strictly in keeping with the drawing, both as to character and simplicity. 

On page 95 a large number of corner designs and borders is presented, one -third of them orig- 
inal designs, by the writer, for this work. The principle of their construction is illustrated by Fig. 
155, in which the larger design shows the necessary preliminary lines, and the smaller the complete 
corner. It is evident in this, as in all cases of interlaced designs, that we must first lay off each 
way from the corner as many equal distances as there are bands and spaces, and lightly make a 
network of squares — or of rhombi, if the angles are acute — by pencilled construction -lines through 
the points of division. 

267. Shade lines on borders. The usual rule as to shade lines applies equally to these designs, 
thus: Following any band or pair of lines making the turns as one 
piece, if it runs horizontally the loiver line is the heavier, while in a 
vertical pair the right-hand line is the shaded line. This is on the 
assumj^tion that the light is coming in the direction usually assumed for 
mechanical drawings, i. e., descending diagonally from left to right. 

In case a pair of lines runs obliquely, the shaded lines may be 
determined b}' a study of their location on the designs of the plate of 
borders. 

It need hardly be said that on any drawing and its title the light 
should be supposed to come from but one direction throughout, and not be 
shifted; and the shaded lines should be located accordingly. This rule is always imperative. 

In drawing for scientific illustration or in art work it is allowable to depart from the usual 
strictty conventional direction of light, if a better effect can thereby be secured. 

268. A striking letter can be made by drawing the shade line only, as in Fig. 146, page 90, which 
we may call "Full -Block Shade -Line," being based upon the alphabet of Fig. 148, page 92, as to 
construction. Owing to its having more projecting parts it gives a much handsomer effect than the 

The student will notice that the light comes from different directions in the two examples. 

These forms are to the ordinary fully -outlined letters what art work of the "impressionist" 
school is to the extremely detailed and painstaking work of many; what is actually seen suggests 
an equal amount not on the paper or canvas. 

269. While a teacher of draughting may well have on hand, as reference works for his class, 
such books on lettering as Prang's, Becker's and others equally elaborate, yet they will be found of 
only occasional service, their designs being as a rule more highly ornate than any but the sijecialist 
would dare undertake, and mainly of a character unsuitable for the usual work of the engineering or 
architectural draughtsman, whose needs were especially in mind when selecting tjqjes for this work. 

The aljjhabets appended afford a large range of choice among the handsomest forms recently 
designed by the leading type manufacturers, also containing the best among former types; and with 
the "Railroad," Full-Block and Half- Block alphabets of this chapter, j^roportioned and drawn by the 
writer, supply the student with a practical "stock in trade" that it is believed will require but 
little, if any, supplementing. 



COPYING PROCESSES. — DRAWING FOR ILLUSTRATION. 97 



CMAPTJEB nil. 

BLUE-PEINT AND OTHER COPYING PROCESSES. — METHODS OF ILLUSTRATION. 

270. While in a draughting office the process described below is, at present, the only method 
of copying drawings with which it is absolutely essential that the draughtsman should be thoroughly 
acquainted, he may, nevertheless, find it to his advantage to know how to prepare drawings for 
reproduction by some of the other methods in most general use. He ought also to be able to 
recognize, usuallj', by a glance at an illustration, the method by which it was obtained. Some brief 
hints on these points are therefore introduced. 

This is, obviously, however, not the place to give full particulars as to all these processes, even 

■ were the methods of manipulation not, in some cases, still " trade secrets " ; but all important details 

concerning them, that have become common property, may be obtained from the following valuable 

works: Modern Hdiographic Processes,* by Ernst Lietze; Photo- Engraving, Etching and Lithography, 'i by 

W. T. Wilkinson; and Modern Reproductive Graphic Processes,'* by Jas. S. Pettit. 

THE BLUE -PRINT PROCESS. 

271. By means of this process, invented by Sir John Herschel, any number of copies of a draw- 
ing can be made, in white lines on a blue ground. In Arts. 43 and 45 some hints will be found 
as to the relative merits of tracing -cloth and "Bond" paper, for the original drawing. 

A sheet of pajDer may be sensitized to the action of light by coating its surface with a solution 
of red prussiate of potash (ferrocyanide of potassium) and a ferric salt. The chemical action of 
light upon this is the production of a ferrous salt from the ferric compound; this combines with 
the ferrocyanide to i^roduce the final blue undertone of the sheet; while the portions of the paper, 
from which the light was intercepted by the lines of the drawing, become white after immersion in 
water. 

The proportions in which the chemicals are to be mixed, are, apparentlj^, a matter of indiffer- 
ence, so great is the disparity between the recipes of different writers ; indeed, one successful 
draughtsman says: "Almost any proportion of chemicals will make blue-prints." Whichever recipe 
is adopted — and a considerable range of choice will be found in this chapter — the hints immediately 
following are of general application. 

272. Any white paper will do for sensitizing that has a hard finish, like that of ledger paper, 
so as not to absorb the chemical solution. 

To sensitize the paper dissolve the ferric salt and the ferrocyanide in water, separately, as they 
are then not sensitive to the action of light. The solutions should be mixed and applied to the 
paper only in a dark room. 

Although there is the highest authority for " floating the paper to be sensitized for two minutes 
on the surface of the liquid " yet the best American practice is to apply the solution with a soft 
flat brush about four inches wide. The main object is to obtain an even coat, which may usually 



* Published by the D. Van Nostrand Company, New York, t American Edition revised and published by Edward L. 
Wilson, New York. 



98 THEORETICAL AND PRACTICAL GRAPHICS. 

be secured by a primary coat of horizontal strokes followed by an overlay of vertical strokes; the 
second coat apjDlied before the first dries. If necessary, another coat of diagonal strokes may be 
given to secure evenness. The thicker the coating given the longer the time required in printing. 
A bowl or flat dish or plate will be found convenient for holding the small portion of the solution 
required for use at any one time. The chemicals should not get on the back of the sheet. 

Each sheet, as coated, should be set in a dark place to dry, either "tacked to a board by two- 
adjacent corners", or "hung on a rack or over a rod", or "placed in a drawer — one sheet in a 
drawer", — varying instructions, illustrating the quite general truth that there are usually several 
almost equally good ways of doing a thing. 

273. To copy a drawing place the prepared paper, sensitized side up, on a drawing-board or 
printing- frame on which there has been fastened, smoothly, either a felt pad or canton flannel cloth. 
The drawing is then immediately placed over the first sheet, inked side up, and contact secured 
between the two by a large sheet of plate glass, jjlaced over all. 

Exposure in the direct rays of the sun for four or five minutes is usually sufficient. The- 
progress of the chemical action can be observed by allowing a corner of the paper to project beyond 
the glass. It has a grayish hue when sufficiently exposed. 

If the sun's rays are not direct, or if the day is cloudy, a proportionately longer time is required,, 
running up in the latter case, from minutes into hours. Only experiment will show whether one's 
solution is " quick " or " slow ; " or the time required by the degree of cloudiness. 

A solution will print more quickly if the amount of water in it be increased or if more iron 
is used ; but in the former case the print will not be as dark, while in the latter the results, as ta 
whiteness of lines, are not so apt to be satisfactory. 

Although fair results can be obtained with paper a month or more after it has been sensitized, 
yet they are far more satisfactory, if the paper is prepared each time (and dried) just before using. 

On taking the j^rint out of the frame it should be immediately immersed and thoroughly 
washed in cold water for from three to ten minutes, after which it may be dried in either of the 
ways previously suggested. 

If many prints are being made, the water should be frequently changed so as not to become- 
charged with the solution. 

274. The entire process, while exceedingly simple in theory, varies, as to its results, with the- 
experience and judgment of the manipulator. To his choice the decision is left between the following 
standard recipes for preparing the sensitizing solution. The " parts " given are all by weight. In. 
every case the potash should be pulverized, to facilitate its dissolving. 

No. 1. (From Le Genie Civil). 

f Red Prussiate of Potash '. 8 parts. 

Solution Nc '' 



No. 1. I 



Water 70 parts. 

f Citrate of Iron and Ammonia 10 parts. 

Solution No. 2. \ ,,, ^ _. 

( Water 70 parts. 

■ Filter the solutions separately, mix equal quantities and then filter again. 
No. 2. (From U. S. Laboratory at Willett's Point). 



Solution No. 1. 
Solution No. 2. 



Double Citrate of Iron and Ammonia 1 ounce. 

Water 4 ounces.. 

Red Prussiate of Potassium ;..... 1 ounce. 

Water 4 ounces. 



Stock Soluti 



BLUE-PRINT PROCESS. 99 

No. 3. (Lietze's Method). 
5 ounces, avoirdupois, Red Prussiate of Potash. 
32 fluid ounces Water. 



" After the red prussiate of potash has been dissolved — which requires from one to two days^ 
the liquid is filtered. This solution remains in good condition for a long time. Whenever it is 
required to sensitize paper, dissolve, for every two hundred and forty square feet of paper 

{1 ounce, avoirdupois. Citrate of Iron and Ammonia, 
4} fluid ounces Water, 

and mix this with an equal volume of the stock solution. 

The reason for making a stock solution of the red prussiate of potash is, that it takes a con- 
siderable time to dissolve and because it must be filtered. There are many impurities in this 
chemical which can be removed by filtering. Without filtering, the solution will not look clear. 
The reason for making no stock solution of the ferric citrate of ammonia is that such solution soon 
becomes moldy and unfit for use. This ferric salt is brought into the market in a very pure state 
and does not need to be filtered after being dissolved. It dissolves very rapidly. In the solid 
form it may be preserved for an unlimited time, if kept in a well -stoppered bottle and protected 
against the moisture of the atmosphere. A solution of this salt, or a mixture of it with the solution 
of red prussiate of potash, will remain in a serviceable condition for a number of days, but it will 
spoil, sooner or later, according to atmospheric conditions. . . . Four ounces of sensitizing solution, 
for blue i:)rints, are amply sufficient for coating one hundred square feet of iDajser, and cost about 
six cents." 

For copying tracings in blue lines or black, on a white ground, one may either employ the 
recipes given in Lietze's and Pettit's works or obtain paper, already sensitized, from the leading dealers 
in draughtsmen's supplies. The latter course has become quite as economical, also, for the ordinary 
blue- print, as the preparing of one's own supply. 

For copying a drawing in any desired color the following method, known as Tilhei''s, is 
said to give good results : " The iiiajjer on which the copy is to ajjpear is first dipjDed in a 
bath consisting of 30 parts of white soap, 30 parts of alum, 40 parts of English glue, 10 j^arts 
of albumen, 2 p)arts of glacial acetic acid, 10 parts of alcohol of 60°, and 500 parts of water. It 
la afterward put into a second bath, which contains' 50 jaarts of burnt umber ground in alcohol, 
20 parts of lampblack, 10 parts of English glue, and 10 parts of bichromate of potash in 500 jjavta 
of water. They are now sensitive to light, and must, therefore, be preserved in the dark. In 
preparing paper to make the positive print another bath is made just like the first one, excei^t that 
lampblack is substituted for the burnt umber. To obtain colored positiA'es the black is replaced by 
some red, blue or other pigment. 

In making the copy the drawing to be copied is put in a p>hotographic printing frame, and the 
negative paper laid on it, and then exposed in the usual manner. In clear weather an illumination 
of two minutes will suffice. After the exposure the negative is put in water to develop it, and the 
drawing will appear in white on a dark ground; in other words, it is a negative or reversed jDicture. 
The paper is then dried and a positive made from it by placing it on the glass of a printing- 
frame and laying the positive paper upon it and exi)osing as before. After placing the fi-ame in 
the sun for two minutes the positive is taken out and put in water. The black dissolves off without 
the necessity of moving back and forth." 



100 THEORETICAL AND PRACTICAL GRAPHICS. 

PHOTO- AND OTHER PROCESSES. 

275. If a drawing is to be reproduced on a different scale from that of the original, some one 
of the processes which admits of the use of the camera is usually employed. Those of most 
importance to the draughtsman are (1) wood engraving; (2) the "wax process" or cerography ; (3) 
lithography, and (4) the various methods in which the photographic negative is made on a film of 
gelatine which is then used directly — to print from, or indirectly — in obtaining a metal plate from 
which the impressions are taken. 

In the first three named above the use of the camera is not invariably an element of the 
process. 

All under the fourth head are essentially photo -processes and their already large number is 
constantly increasing. Among them may be mentioned photogravure, collotype, phototype, autotype, photo- 
glyph, alberiype, heliotype, and heliogravure. 

W'OOD ENGRAVING. 

276. There is probably no process that surpasses the best work of skilled engravers on wood. 
This statement will be sustained by a glance at Figs. 14, 15, 20-24, 134, 136, and those illustrating 
mathematical surfaces, in the next chapter. Its expensiveness and the time required to make an 
illustration by this method are its only disadvantages. 

Although the camera is often employed to transfer the drawing to the boxwood block in which 
the lines are to be cut, yet the original drawing is quite as frequently made in reverse, directly on 
the block, by a professional draughtsman who is supposed to have at his disposal either the object 
to be drawn or a j)hotograph or drawing thereof The outlines are pencilled on the block and the 
shades and shadows given in brush tints of India ink, re -enforced, in some cases, by the pencil, for 
the deepest shadows. 

The "high lights" are brought out by Chinese white. A medium wash of the latter is also 
usually spread upon the block as a general preliminary to outlining and shading. 

The task of the engraver is to reproduce faithfully the most delicate as well as the strongest 
effects obtained on the block with pencil and brush, cutting away all that is not to appear in black 
in the print. The finished block may then be used to print from directly, or an electrotype block 
can be obtained from it which will stand a large number of impressions much better than the wood. 

CEROGRAPHY. 

277. For map -making, illustrations of machinery, geometrical diagrams and all work mainly in 
straight lines or simple curves, and not involving too delicate gradations, the cerographic or " wax 
process" is much employed. For clearness it is scarcely surpassed by steel engraving. Figures 36, 
90 and 107 are good specimens of the effects obtainable by this method. The successive steps in the 
process are (a) the laying of a thin, even coat of wax over a copper plate; (b) the transfer of 
the drawing to the surface of the wax, either by tracing or — more generally — by photography; (c) 
the re -drawing or rather the cutting of these lines in the wax, the stjdus removing the latter to the 
surface of the copper; (d) the taking of an electrotj^pe from the plate and wax, the deposit of 
copper filling in the lines from which the wax was removed. 

Although in the preparation of the original drawing the lines may preferably be inked yet this 
is not absolutely necessary, provided a pencil of medium grade be employed. 



LITHOGRAPHY.— PHOTO-ENGRAVING. 101 

Any letters desired on the final plate may be also pencilled in their jDroper places, as the 
engraver makes them on the wax with type. 

A surface on which section -lining or cross-hatching is desired may have that fact indicated upon 
it in writing, the direction and number of lines to the inch being given. Such work is then done 
with a ruling machine. 

Errors may readilj^ be corrected, as the surface of the wax may be made smooth, for recutting, 
by passing a hot iron over it. 

LITHOGRAPHY. — PHOTO - LITHOGEAPHY. — CHROJIO - LITHOGRAPHY. 

278. For the lithographic process a fine-grained, imported limestone is used. The drawing is 
made with a greasy ink — known as " lithograiahic " — upon a specially prepared paper, from which 
it is transferred, under pressure, to the surface of the stone. The un- inked parts of the stone are 
kept thoroughly moistened with water, which j^revents the printer's ink (owing to the grease which 
the la/tter contains) from adhering to any jjortion except that from which the impressions are desired. 

Photo -lithography is simply lithography, with the camera as an adjunct. The positive might be 
made directly upon the surface of the stone by coating the latter with a sensitizing solution ; but, 
in general, for convenience, a sensitized gelatine film is exjiosed under the negative, and by subsec[uent 
treatment gives an image in relief which, after inking, can be transferred to the surface of the stone 
as in the ordinary process. 

Chromo- lithography, or lithography in colors, has been a very expensive process owing to its 
requiring a separate stone for each color. Recent inventions render it probable that it will be much 
simplified and the expense correspondingly reduced. The details of manipulation are closely analogous 
to those for ink prints. 

When colored plates are wanted, in which delicate gradations shall be indicated, chromo -lithography 
may preferably be adopted ; although " half tones," with colored inks, give a scarcely less pleasing 
eifect, as illustrated by Figs. 7-10, Plate II. But for simple line -work, in two or more colors, one 
may preferably employ either cerography or jDhoto - engraving, each of which has not only an 
advantage, as to expense, over any lithograj)hic process, but also this in addition — that the blocks 
can be used b}' anj' jjrinter; whereas lithographing establishments necessarily not only prepare the 
stone but also do the printing. 

PHOTO - ENGRAVING. — PHOTO - ZINCOGRAPHY. 

279. In this popular and rapid process a sensitized solution is spread ui^on a smooth sheet of 
zinc and over this the photographic negative is placed. Where not acted on by the light the 
coating remains soluble and is washed away, exposing the metal, which is then further acted on by 
acids to give more relief to the remaining portions. 

Except as described in Art. 281 this process is only adapted to inked work in lines or dots, 
which it reproduces faithfully, to the smallest detail. Among the best photo - engravings in this book 
are Figs. 12, 13, 50, 79 and 80. 

280. The following instructions for the preparation of drawings, for reproduction by this process, 
are those of the American Society of Mechanical Engineers as to the illustration of papers by its 
members, and are, in general, such as all the engraving companies furnish on application. 

"All lines, letters and figures must be perfectly black on a white ground. Blue prints are not 
available, and red figures and hues will not appear. The smoother the paper, and the blacker the 
ink, the better are the results. Tracing- cloth or paper answers very well, but rough paper — even 



102 THEORETICAL AND PRACTICAL GRAPHICS. 

Whatman's — gives bad lines. India ink, ground or in solution, should be used; and the best lines 
are made on Bristol board, or its equivalent with an enameled surface. Brush work, in tint or 
grading, unfits a drawing for immediate use, since only line work can be photographed. Hatching 
for sections need not be completed in the originals, as it can be done easily by machine on the 
block. If draughtsmen will indicate their sections unmistakably, the}' will be properly lined, and 
tints and shadows will be similarly treated. 

The best results may be expected by using an original twice the height and width of the 
proposed block. The reduction can be greater, provided care has been taken to have the lines far 
enough aj)art, so as not to mass them together. Lines in the plate may run from 70 to 100 to 
the inch, and there should be but half as many in a" drawing which is to be reduced one half; 
other reductions will be in like proportion. 

Draughtsmen may use photographic prints from the objects if they will go over with a carbon 
ink all the lines which they wish reproduced. The photographic color can be bleached away by 
flowing a solution of bi- chloride of mercury in alcohol over the print, leaving the pen lines only. 
Use half an ounce of the salt to a pint of alcohol. 

Finally, lettering and figures are most satisfactorily printed from tyjje. Draughtsmen's best efforts 
are usually thus excelled. Such letters and figures had therefore best be left in pencil on the 
drawings, so they will not photograph but may serve to show what type should be inserted." 

To the above hints should be added a caution as to the use of the rubber. It is likely to 
diminish the intensity of lines already made and to affect their sharpness ; also to make it more , 
difficult to draw clear-cut lines wherever it has been used. 

It may be remarked with regard to the foregoing instructions that they aim at securing that 
uniformity, as to general appearance, which is usually quite an object in illustration. But where the 
preservation of the individuality and general characteristics of one's work is of any importance what- 
ever, the draughtsman is advised to letter his own drawings and in fact finish them entirely, himself, 
with, perhaps, the single exception of section -lining, which may be quickly done by means of Day^s 
Rapid Shading Mediums or by other technical processes. 

281. Half Tones. Photo -zincography may be employed for reproducing delicate gradations of light 
and shade, by breaking up the latter when making the ijhotographic negative. The result is called 
a half tone and it is one of the favorite jarocesses for high-grade illustration. Figs. 95 and 130 
illustrate the effects it gives. On close inspection a series of fine dots in regular order . will be 
noticed, so that no tone exists unbroken, but all have more or less white in them. 

The methods of breaking up a tone are very numerous. The first pateiat dates back to 1852. 
The i^rinciple is practically the same in all, viz., between the object to be photographed and the 
plate on which the negative is to made there is interjjosed a " screen " or sheet of thin glass on 
which the desired mesh has been previously photograjDhecl. 

In the making of the "screen" lies the main difference between the variously - named methods. 
In Meissenbach's method, by which Figs. 95 and 130 were made, a photograph is first taken, on the 
"screen," of a pane of clear glass in which a system of parallel lines — one hundred and fifty to the 
inch — has been cut with a diamond. The ruled glass is then turned at right angles to its first 
position and its lines photographed on the screen over the first set, the times of exposure differing 
slightly in the two cases, being generally about as 2 to 3. 

This process is well adapted to the reproduction of "wash" or brush -tinted drawings, photographs, 
etc. The object to be represented, if small, may preferably be furnished to the engraving company 
and they will photograph it direct. 



PHOTOGRAPHIC ILLUSTRATIVE PROCESSES. 103 

GELATINE FILM PHOTO - PROCESSES. 

282. As stated in Art. 275, in which a few of tlie above processes are named, a gelatine film 
may be employed, either as an adjunct in a method resulting in a metal block, or to print from 
directly ; in the latter case the prints must be made, on special paper, by the company prei^aring 
the film. In the composition and manipulation of the film lies the main difference between otherwise 
closely analogous processes. For any of them the comiDany should be supplied with either the original 
object or a good drawing or photographic negative thereof 

Not to unduly prolong this chapter — which any intelligible distinction between the various, 
methods would involve, yet to give an idea of the general principles of a gelatine process I 
conclude with the details of the preparation of a heliotype plate, given in the language of one 
company's circular. Figs. 1 — 5 of Plate "II illustrate the effect obtained by it. 

" Ordinary cooking gelatine forms the basis of the jDositive plate, the other ingredients being bichromate 
of potash and chrome alum. It is a peculiarity of gelatine, in its normal condition, that it will absorb 
cold water, and swell or exjiand under its influence, but that it will dissolve in hot water. In the prej)ara- 
tion of the plate, therefore, the three ingredients just named, being combined in suitable iiroportions, 
are dissolved in hot water, and the solution is poured upon a level jDlate of glass or metal, and left 
there to dry. When dry it is about as thick as an ordinary sheet of parchment, and is strij^ped 
from the drying-plate, and placed in contact with the previously-prepared negative, and the two 
together are exi^osed to the light. The presence of the bichromate of potash renders the gelatine 
sheet sensitive to the action of light; and wherever light reaches it, the plate, which was at first 
gelatinous or absorbent of water, becomes leathery or waterproof In other words, wherever light 
reaches the plate, it produces in it a change similar to that which tanning produces upon hides in 
converting them into leather. Now it must be understood that the negative is made up of trans- 
parent parts and opaque parts ; the transparent parts admitting the passage of light through them, 
and the opaque parts excluding it. When the gelatine f)late and the negative are placed in contact, 
they are exposed to light with the negative uppermost, so that the light acts through the translucent 
portions, and waterproofs the gelatine underneath them ; while the opaque portions of the negative 
shield the gelatine underneath them from the light, and consequently those parts of the plate remain 
unaltered in character. The result is a thin, flexible sheet of gelatine, of which a jDortion is water- 
proofed, and the other portion is absorbent of water, the waterproofed jjortion being the image which 
we wish to rejDroduce. Now we all know the repulsion which exists between water and any form 
of grease. Printer's ink is merely grease united with a coloring-matter. It follows, that our gelatine 
sheet, having water applied to it, will absorb the water in its unchanged parts; and, if ink is then 
rolled over it, the ink will adhere only to the waterproofed or altered parts. This flexible sheet of 
gelatine, then, prepared as we have seen, and having had the image impressed upon it, becomes the 
heliotype plate, capable of being attached to the bed of an ordinary i^rinting - press, and printed in the 
ordinary manner. Of course, such a sheet must have a solid base given to it, which will hold it 
firmly on the bed of the press while printing. This is accomjDlished by uniting it, under water, with 
a metallic plate, exhausting the air between the two surfaces, and attaching them by atmospheric 
pressure. The plate, with the printing surface of gelatine attached, is then placed on an ordinary 
platen printing-press, and inked up with ordinary ink. A mask of paper is used to secure white 
margins for the prints; and the impression is then made, and is ready for issue." 



" The study of Descriptive Geometry possesses an important philosophical peculiarity, quite 
independent of its high industrial utility. This is the advantage which it so pre-eminently 
offers in habituating the mind to consider very complicated geometrical combinations in space, 
and to follow with precision their continual correspondence with the figures which are actually 
traced — of thus exercising to the utmost, in the most certain and precise manner, that im- 
portant faculty of the human mind which is properly called ' imagination,' and tohich consists, 
in its elementary and positive acceptation, in rejiresenting to ourselves, clearly and easily, a 

vast and variable collection of ideal objects, as if they were really before us While it 

belongs to the geoinetry of the ancients by the character of its solutions, on the other hand 
it appiroaches the geometry of the moderns by the nature of the questions which compose it. 
These questions are in fact eminently remarkable for that generality which constitutes the true 
fundamental character of tnodern geometry; for the methods used are always conceived as 
apjjlicable to any figures whatever, the peculiarity of each having only a purely seco?ida?-y 
influence." Acguste Comte : Cours de PMlosophie Positive. 

" A mathematical problein may usually be attacked by lohat is terined in military par- 
lance the method of ' systematic approach,;' that is to say, its solution may be gradually felt 
for, even though the successive steps leading to that solution cannot be clearly foreseen. But 
a Descriptive Geometry problem must be seen through and through before it can be attempted. 
The entire scope of its conditions as well as each step toioard its solution must be grasped 
by the imagination. It must be ' taken by assault. ' ' ' 

Geoege Sydenham Clarke, CaiJtain, Boyal Engineers. 



MECHANICAL DRAWINGS.— WORKING DRAWINGS. 131 



CHAPTEB X. 

PROJECTIONS AND INTERSECTIONS BY THE THIRD- ANGLE . METHOD.— THE DEVELOPMENT OF 

SURFACES FOR SHEET METAL PATTERN MAKING. — PROJECTIONS, INTERSECTIONS AND 

TAN6ENCIES OF DEVELOPABLE, WARPED AND DOUBLE CURVED 

SURFACES, BY THE FIRST -ANGLE METHOD. 

383. The me(*hanical drawings ]ireliminary \o the construction of machinery, blast furnaces, stone 
arclics. buildincrs, and, in fact, all architectural and engineering projects, are made in accordance with 
the ])rindplcs of Descripti^-e Geometry, ^^'llen fully dimensioned the}' are called working drawings. 

The oliject to be reiiresented is supposed to be jalaced in either the first or the third of the 
four angles formed by the intersection of a horizontal plane, H, with a vertical plane, Y. (Fig. '228). 

The representations of the object upon the planes are, in mathematical language, ■projections,* and 
are iilitained liy drawing perpendiculars to the planes H and V from the various points of the 
object, the jioint of intersection of each such projecting line with a plane gi^'ing a projection of the 
original point. Such drawings are, obviously, not ''views" in the ordinary sense, as the)' lack the 
perspective effect which is involved in having the j^oint of sight at a finite distance; yet in ordinary 
parlance the terms top vieii:, horizontal projection and plan are used synonymously; as are front view 
and front elevation with vertical projection, and side elevation with profile view, tlie latter on a j)lane 
perpendicular to both H and V and called the profile plane. 

^ Until the last decade of the first century of Descriptive Geometry (1795-1895) problems were 
solved as far as j^ossible in the first angle. As the location of the oliject in tlie third angle — that 
is, below the horizontal plane and behind the vertical — results in a grouping of the views which is 
in a measure self-interpreting, the Third Angle Method is, however, to a considerable degree sujiplant- 
ing the other for machine-shop work. 

The advantageous grouping of the j^i'^^jections which constitutes the only — though a cpiite suf- 
ficient — justification for giving it special treatment, is this: The front view being alwaj's the central 
one of the grouji, the top view is found at the top; the view of the right side of the object appears 
on the right; of the left-hand side on tlie left. etc. 'Thus, in Fig. 228 (a), with the hollow block 
BDFS as the object to be represented, we have odes for its horizontal jjrojection, r'd'e'f for its 
vertical projection, f"e"s".v" for the side elevation; then on rotating the plane H clockwise on G. L. 
into coincidence with Y, and the profile jilane P about Q R until the pvo^ection- f" e" s" x" reaches 
f"'e"'s"'.r"', we would have tliat location of the views which has just been described. 

The lettering shows that each projectiim re}iresents that side of the oliject whicli is toward the 
plane of projection. 

384. The same grouj)ing can Ije arrived at liy a different conception, which will, to some, have 
advantages over the other. It is illustrated l\v Fig. 228 (b), in which the same oliject as before is 



*For the convenience of those who have to take up this subject without previous study of Descriptive Geometry the 
Third-Angle section of tliis chapter is made complete in itself, hy the re-statement of the principles involved and which have 
been treated at somewhat greater length in the previous chapter: although a review of such luatter may be by no means 
disadvantageous to those who have already been over the fundamentals. 



132 



THEORETICAL AND PRACTICAL GRAPHICS. 



sujoposed to be smTdumled by a system of mutually -peqiendieular transparent ]ilanes, or, in other 
words, to be in a box having glass sides., and on each side a drawing made of what is seen through 
that side, excluding the idea, as before, of jierspective ^new, anil representing each point by a per- 

X^ig-- 233, 




iiPiii||l||[r|[i(B«p™R|^ 



pendicular from it to the j)lane. The whole sj^stem of box and planes, in the wood -cut, is rotated 
90° from the position shown in Fig. 228(a), liringing tliem into the usual position, in which' the 
observer is looking perpendicularly toward the vertical plane. 



^"igr- 22Q. 





385. In Fig. 229 we may illustrate either the First or the Third Angle method, as to the toi> 
view of the object; ndes in the upper plane being the plan hy the latter method, and a^d^e^s^ 
by the former. 



nRTnnnRArHTc projectiox of solids. 



133 



Disregarding QTXX we have the object and planes iUustrating the first -angle method through- 
out, the lettering of each ])rt}jection showing that it represents the side of the object farthest from the 
plane, making it the exact reverse of the third- angle sj'stem. 

In tlic ordiiinri/ representation the same object -n'ould be reiDresented simply by its three views as 
in Fig. 280. In the elevations the short -dash Hnes indicate the in^^sible edges of the hole. 

The arcs show the rotation which carries the profile view into its proper place. 



231. 







Fig. 3. 


'r- 
e" i 

Ac 
















I 1 






C" 


Fig. 8. 

t 


r^L . 






-i_ 






'■'Ic: 



386. For the sake of more readily contrasting the two methods a group of views is shown in 
Fig. 231, all above G. L., illustrating an object by the First Angle system, while all below HK 
represents the same oliject by the Third Angle method. 

When looking at Figures 1, 2, 3 and 4 the observer queries: AVhat is the object, in space, whose 
front is like Fig. 1, top is like Fig. 2, left side is Hke Fig. 3 and riciht side like Fig. 4? 

For the view of the left side he might imagine himself as having Ijeen at first between G and 
H. looking in the direction of arrow X, after which both himself and the object were turned, together 



134 THEORETICAL AND PRACTICAL GRAPHICS. 

to the right, through a ninety -degree arc, when the same side would be presented to his view 
in Fig. 3. Similarly, looking in the direction of the arrow M, an eqiial rotation to the left, as 
indicated by the arcs 1-2, 3-4, 5-6, etc., would give in Fig. 4 the view obtained from direction 
M. His mental queries would then be answered about as follows: Evidently a cubical block with 
a rectangular recess — r'v'd'c' — in front; on the rear a prismatic projection, of thickness p /( and 
whose height equals that of the cube; a short cylindrical ring projecting from the right face of the 
cube; an angular j^rojecting piece on the left face. 

In Fig. 2 the line rv is in short dashes, as in that view the back plane of the recess r'v'd'c' 
would be invisible. In Fig. 4 the back plane of the same recess is given the letters, v"d", of the 
edge nearest the oliserver from direction M. 

To illustrate the third angle method by Fig. 231 we ignore all above the line H K. In Fig. 5 we 
have the same front elevation as before, but above it the view of the top; below it the view of the 
bottom exactly as it would appear were the object held before one as in Fig. 5, then given a ninety- 
degree turn, around a'b', until the under side became the front elevation. 

Fig. 7 may as readilj' be imagined to be obtained by a shifting of the object as by the rotation 
of a plane of projection; for bj' translating the object to the right, from its jDOsition in Fig. 5, then 
rotating it to the left 90° about b'n', its right side would appear as shown. 

387. For convenient reference a general resume of terms, abbreviations and instructions is next 
presented, once for all, for use in both the Third jungle and First Angle methods. 

(1) H, V, P the horizontal, verftcnl and profile planes of projection respectively. 

(2) H - projector the projecting line which gives the horizontal projection of a point. 

(3) V- projector the projecting line giving the projection of a point on V. 

(4) Projector -plane the profile plane containing the projectors of a point. 

(5) h. p the horizontal projection or plan of a point or figure. 

(6) v. p the vertical projection or elevation of a point or figure. 

(7) h. t horizo7ital trace, the intei'section of a line or surface with H. 

(8) V. t vertical trace, the intersection of a line or surface with V. 

(9) H- traces, V- traces plural of horizontal and vertical traces respectively. 

(10) G-. L ground line, the line of intersection of V and H. 

(11) V-parallel a line parallel to V and lying in a given plane. 

(12) A horizontal any horizontal line lying in a given plane. 

(13) Line of declivity the steepest line, with respect to one plane, that can lie in another plane. 

(14) Eabatment revolution into H or V about an axis in such plane. 

(15) Counter -rabatmenl or revolution . restoration to original position. 

388. For Problems relating solely to the Point, Line and Plane. 

Given lines should be fine, continuous, black ; required liyies heavy, continuous, black or red ; construction lines in 
fine, continuous red, or short-dash black; traces of an auxiliary pdave, or invisible traces of any plane, in dash - and ■ 

For Problems relatinq to Solid Objects. 

(1) Pencilliiir/. Exact ; generally completed for the whole drawing before any inking is done ; the work usually from 
centre lines, and from the larger — and nearer — pai-ts of the object to the smaller or more remote. 

(2) Inlcinr) of the Object. Curves to be drawn before their tangents ; fine lines uniform and drawn before the shade 
lines; shade lines next and with one setting of the pen, to ensvire uniformity. On tapering shade lines see Art. 111. 

(3) Shade Lines. In architectural work these would be drawn in accordance with a given direction of light. 

In American machine-shop practice the right-hand and lower edges of a plane surface are made shade lines if they 
separate it from invisible surfaces. Indicate curvature by line-shading if not otherwise sufficiently evident. (See Fig. 288). 



THE CONSTRUCTION AND FINISH OF WORKING DRAWINGS. 



135 



(4) Invisible lines of the objecf, black, invariably, in dashes nearly one-tenth of an inch in length. 

(5) Inking of lines other than of the object. "When no colors are to be employed the following directions as to 
kind of line are those most frequently made. The lines may preferably be drawn in the order mentioned. 

Centre lines, an alternation of dash and two dots. 

Dimension lines, a dash and dot alternately, with opening left for the dimension. 

Extension lines, for dimensions placed outside the views, in dash-and-dot as for a dimension line. 

Ground line, (when it cannot be advantageously omitted) a continuous heavy line. 

Construction and other explanatory lines in short dashes. 

(6) ]i'hen using colors the centre, dimension and extension lines may be fine, continuous, red; or the former may 
be blue, if preferred. Constructioti lines may also be red, in short dashes or in fine continuous lines. 

Instead of using bottled inks the carmine and blue may preferably be taken directly from Winsor and Newton cakes, 
"moist colors." Ink ground from the cake is also preferable to bottled ink. 

Drawings of developable and warped surfaces are much more effective if their elements are drawn in some color. 

(7) Dimensions and Arrow-Tips. The dimensions should invariably be in black, printed free-hand with a writing- 
pen, and should rend in line with the dimension line they are on. On the drawing as a whole the dimensions should 
read either from the bottom or right-hand side. Fractions should have a horizontal dividing line; although there is 
high sanction for the omission of the dividing line, particularly in a mixed number. 

Extended Gothic, Koman, Italic Roman and Reinhardt's form of Condensed Italic Gothic are the best and most 
generally used types for dimensioning. 

The arrow- tips are to be always drawn free-hand, in black; to touch the lines between which they give a distance; 
and to make an acute instead of a right angle at their point. 

389. Working drawing of a right pyramid; base, an equilateral triangle 0.9" on a side; altitude, x. 

Draw first the equilateral triangle ah c for the plan of the base, making its sides of the pre- 
scribed length. If we make the edge a b perpendicular to ^^s- sss. 
the profile plane, 1, the face v a b will then appear in 
profile view as the straight line v"b". Being a right pyra- 
mid, with a regular base, we shall find v, the plan of the 
vertex, equally distant from a, b and c; and v a, vb, vc 
for the plans of the edges. 

Parallel to G. L. and at a distance apart equal to the 
assigned height, x, draw mv" and no" as upper and lower 
limits of the front and side elevations; then, as the h. p. 
and V. p. of a point are alwaj^s in the same perijendicu- 
lar to G. L., we project r, a, b and c to their respective 
levels by the construction lines shown, obtaining v'.a'b'c' 
for the front elevation. 

Projectors to the profile plane from the points of the plan give 1, 2, 3, which are then carried, 
in arcs about 0, to L, 5, 4, and i)rojected to their proper levels, giving the .s/V/e deration, v"b"c". 

As the actual length of an edge is not shown in either of the three views, we employ the fol- 




E'ig-. 233- 




is the plane area which, folded 



on 



lowing construction to ascertain it: Draw vv^ ijeri^endicular to 
vb, and make it equal to x; 'v,b is then the real length of 
the edge, shown by rabatment about v b. 

The develojmient of the pyramid (Fig. '233) may be obtained 
by drawing an arc ABCA^ of radius = Ui 6 (the true length 
of edge, from Fig. 232) and on it laying off the chords A B, 
B C, CA^ equal to ah, he, ca of the plan; then V-ABCA^ 
VC and V B, wi.iuld give a model of the pyramid represented. 



im 



THEORETICAL AND PRACTICAL GRAPHICS. 



390. Wm-klng drawing of n semi -cylindrical pij)e: outer diameter, x; 



inner diameter, 



y; 



height, 



For the plan draw concentric semi -circles aed and b s c, of diameters 
ing their extremities by straight lines a b, c d. At a distance 
apart of z inches draw the upper and lower limits of the 
elevations, and project to these levels from the points of the 
2)lan. 

In the side view the thickness of the shell of the cylinder 
is shown liy the distance between e"f" and s"i" — the latter so 
drawn as to indicate an invisible limit or line of the object. 

The lino shading would usually be omitted, the shade lines 
generally sufficing to convey a clear idea of the fomi. 

391. Half of a hollow, hexagonal prism. In a semi-circle of 
diameter ad step off the radius three times as a chord, giving 
the vertices of the plan a h c d of the outer surface. Parallel to 
b c and at a distance from it equal to the assigned thickness 
of the prism, draw ef temiinating it on lines (not shown) 



and y resj)ectively, join- 




^ig-. 335. 




From e and / draw e h 
ccl. Drawing a' c" and 



drawn through b and c at 60 ° to a d. 

and fg, parallel respectively to a b and 

7n't" as upper and lower limits, f)roject to them as in preceding 

problems for the front and side elevations. 

392. Working drawing of a hollow, prismatic block, standing 
obliquely to the vertical and profile planes. 

Let the block be 2"x3"xl" outside, with a square oijcn- 
ing 1 " X 1 " X 1 " through it in the direction of its thickness. 
Assuming that it has been required that the two -inch edges 
should be vertical, we first draw, in Fig. 236, the jilan asxb, 
3"xl", on a scale of 1:2. The inch -wide opening through the 
centre is indicated by the short- dash lines. 

For the elevations the upper and lower limits are drawn 2" 

^igr- a3S. 



apart, and o, b, s, x, etc., projected to them. The 
elevations of the opening are between levels m'm" 
and k'k", one inch apart and equi-distant from 
the upper and lower outlines of the views. The 
dotted construction lines and the lettering will 
enabile the student to recognize the three views 
of any point without difficult}'. 

393. In Fig. 237 we have the same object as 
that illustrated by Fig. 236, but now represented 
as cvit by a vertical j^lane whose horizontal trace 
is V y. Tlie parts of the Itlock that are actually 
cut by the jjlane are shown in section -lines in 
the elevations. This is done here and in some 
later examples merely to aid the beginner in 
understanding the views; but, in engineering prac- 
tice, section -lining is rarely done on views not perpendicida/r to the section plane.. 



3C 


fV-— 


.... 










c 


\V...A. ■ 


^ 


^^^^ 


%± 


--,. 


\ 1 \ 


yv^ 




: 1 


\ \ \ \ \ \ \ 
\ ' ' 1 1 1 ' 




,y^ 


^ 




-., 


v^ l" j 1 


i ' L 




1 i ' ! 
b 'a 1 1 : 


\x 


,s' 


\(l 1 ^! 1 1 is" 1 






cj 


^ 




111 1 


] 










m" I 


£_j. 


e' 




























d 




'L._ 




h 1 i 


t. 



PROJECTION OF S OLIDS. — WORKING DRAWINGS. 



137 



it is customaiy to omit the 

X-lg-. 237. 




SECTIONAL VIEW 



394. Sujypression of the Ground Line. In machine drawing 
ground line, since tlie fonni< of the various views — 
which alone concern us — are independent of the 
distance of the object from an imaginary horizon- 
tal or vertical plane. We have only to remember 
that all elevations of a point are at the same 
level; and that if a ground line or trace of any 
vertical plane is wanted, it will be perpendicular 
to the line joining the plan of a point with its 
projection on such vertical plane. 

395. Sections. Sectional Views. Although earlier 
defined (Art. 70), a re-statement of the distinction 
between these terms may well precede problems in 
which they will be so frequently employed. 

When a plane cuts a solid, that portion of the 
latter which comes in actual contort with the cutting 
plane is called the .section. 

A sectional view is a view perpendicular to the cutting plane, and showing not only the section liut also 
the object itself as if seen through 
the plane. When the cutting 
plaiie is vertical such a view is 
called a sectioned elevation; when 
horizontal, a sectioned plan. 

396. Working clraicing of a 
regidar, pentagoncd pyramid, hollow, 
truncated by an oblique plane; cdso 
the development, or "poHfcn," of the 
outer surface below the cutting plane. 
For data take the aUitude at 2"; 
inclination of faces, 6° (meaning 
any arl)itrary angle); inclinatiim 
of section plane, 30°; distance 
between inner and outer faces of 
pyramid, ^". 

(1) Locate v and v' (Fig. 238) 
for the plan and elevation of the 
vei-tex, taking them sufficiently 
apart to avoid the overlapping of 
one view upon the other. Through 
V draw the horizontal line .S' T, 
regarding it not only as a centre 
line for the plan but also as the 
h. t. of a central, vertical, refer- 
ence plane, parallel to the ordi- 
nary vertical jilane of projection. 




138 



THEORETICAL AND PRACTICAL GRAPHICS. 



SECTIONAL VfEW 



(The student should note that for convenience Fig. 238 is repeated on this page.) 

On the vertical line w' (at first indefinite in length) lay ofT v' s' equal to 2", for the altitude 
(and axis) of the pj'raniid, and through s' draw an indefinite horizontal hne, which will contain the 
V. p. of the base, in both front and side views. 

Draw v'h' at 6° to the horizontal. It will represent the v. p. of an outer face of the pyramid,, 
and b' will be the v. p. of the edge ah of the ham. The base abcde is then a regular pentagon 
circumscribed about a circle of centre v and radius y i = .s' 6'. Since the angle avb is 72° (Art. 92) 
we get a starting corner, a or b, by drawing va or vb at 36° to ST, to intercept the vertical through 
//. The plans of the edges of the pyramid are then v a, vb, v c, vd and v e. Project d to d' and 
draw v' d' for the elevation of 
(' d ; similarly for v e and v e, 
which hajipen in this case to co- 
incide in vertical projection. 

For the inner surface of the 
pyramid, whose faces are at a 
perpendicular distance of \" from 
the outer, begin by drawing g' V 
parallel to and \" fi-om the face 
projected in b' v' ; this will cut 
the axis at a p(jint t' wliieh will 
be the vertex of the inner sur- 
face, and g' t' will represent the 
ele\-ation of the inner face that is 
parallel to the face avb — v'b'; 
while gh, vertically above <"/' and 
included between va and vb, will 
be the plan of the lower edge of 
this face. Complete the pentagon 
gh---k for the jjlan of the inner 
base; project the corners to b' cV 
and join with t' to get the ele- 
vations of the interior edges. 

The. Section. In our figure let 
(?' H' be the section plane, sit- 
uated perpendicular to the ver- 
tical plane and inclined 30° to 
the horizontal. It intersects v' d' in j>', which projects upon vd at p. Similarly, since G' H' cuts 
the edges v' c' and v' e' at points projected in o', we project from the latter to vc and ve, obtaining 
and q. A like construction gives m and n. The polygon mnopq is then the plan of the outer 
boundary of the section. 

The inner edge g' t' is cut by the section plane at /', wliich projects to both vh and vg, giving 
the parallel to mn through /. The inner boundary of the section may then be completed either 
by determining all its vertices in the same way or on the principle that its sides will be parallel to 
those of the outer polygon, since any two planes are cut by a third in parallel lines. 

The line vi' p' is the vertical projection of the entire section. 




PROJECTION OF S OLIDS.— WORKING DRAWINGS. 139 

(2) The side elevation. This might be obtained exactlj- as in the five preceding figures, that is, 
hy actually locating the side vertical, or lyrofile, plane, projecting ujion it and rotating through an 
arc of 90°. In engineering practice, however, the method now to be described is in far more general 
use. It does not do away with the profile plane, on the contrary presupposes its existence, but 
instead of actually locating it and drawing the arcs which so far have kejDt the relation of the views 
■constantly before the eye, it reaches the same result in the following manner : A vertical line 6" T' 
is drawn at some convenient distance to the right of the front elevation ; the distance, from iS T, of 
any point of the plan, is then laid off horizontally from S' T', at the same height as the front 
■elevation of the point. For, as earlier stated, S T was to be regarded as the horizontal trace of a 
vertical plane. Such plane would, e-^-idently, cut a profile plane in a vertical line, whicli we may 
•call S'T', and let the S' T' of our figure represent it after a ninety - degree rotation has occurred. 
The distances of all points of the object, to either the front or rear of the vei-tical plane on S T, 
would, obviously, he now seen as distances to the left or right, respectively, of the trace S' T', and 
would be directly transferred with the dividers to the lines indicating their level. Thus, e" is on 
the level of e', but is to the right of S'T' the same distance that e is above (or, in reality, behind) 
the plane ST; that is, e" d" equals eu. Similarly d"b" equals ib; n"x" equals nx. 

It is usual, wliere the object is at all s.ymmetrical, to locate these reference planes centrally, so 
that tlieir traces, used as indicated, ma}' bisect as many lines as possible, to make one setting of the 
•di\'iclers do double work. 

(3) T)-ve size of the Section. Sectioned View. If the section plane G' H' were rotated directly 
about its trace on the central, vertical plane S T, until j^arallel to the paper, it would show the 
.section m'p' — mnopq in its true size; but such a construction would cause a confusion of lines, 
the new figure overlapping the front elevation. If, however, we transfer the plane G' H' — keeping 
it parallel to its first position during the motion — to some new position S"T'', and then turn it 90° 
•on that line, we get m,7i,o,p, g,, the desired Adew of the section. The distances of the vertices 
of the section from S" T" are derived from reference to S T exactly as were those in the side 
■elevation; that is, v} ^x^ = mx = m"x". We thus see that one central, vertical, reference plane ST is 
auxihary to the construction of two impoi-tant views ; S' T' represents its intersection with the profile 
■or side vertical plane, while S" T" is its (transferred) trace upon the section plane G'H'. For the 
remainder of the sectional views the points are obtained exactly as above described for the section; 
thus c'c^c, is perpendicular to S" T" ; e,ii^ equals eu, and c,?'-! equals cv. 

(4) To determine the actual length of the vanous edges. The only edge of the original, uncut 
pyramid, that would require no construction in order to show its true length, is the extreme right- 
hand one, which — being parallel to the vertical plane, as shown by its plan vd being horizontal — is 
seen in elevation in its true size, v'd'. Since, however, all the edges of the pj'ramid are equal, we 
maj^ find on r'd' the true length of any portion f)f some other edge, as, for examjale o'c', b}^ taking 
that part of r'd' which is intercepted between the same horizontals, viz.: o"'d'. 

Were we compelled to find the true length of o'c', oc, independently of any such convenient 
relation as that just indicated, we would apply one of the methods fully illustrated hj Figs. 183, 184 
and 187, or tlie follomng "shop" modification of one of them: Parallel to the plan oc draw a line 
y z, their distance apart to be equal to the difference of level of o' and c', ■which difference may be 
■obtained frf)m eitlier of the elevations. From the plan o of tlie higher end of the line draw the 
-common perpendicular of and join f with c, obtaining the desired length fc. 

(5) To shoiv the exact form of any face of the pyramid. Taking, for example, the face o c d pj, 
revolve o j) about the horizontal edge cd until it reaches the level of the latter. The actual distance 



140 



THEORETICAL AND PRACTICAL GRAPHICS. 



of from c, and of p from d will be the same after as before this revolution, while the paths of o 
and p during rotation will be projected in lines or and pio, each perpendicular to cd; therefore, 
with c as a centre, cut the j^erpendicular or by an arc of radius fc — just ascertained to be the real 
length of oc, and, similarly, cut p%o by an arc of radius du) = 'p'd'; join r with c, iv with d, 
draw 10 r and we have in cdwr the form desired. 

(6) The development of the outer surface of the truncated pyramid. With any point F as a centre 
(Fig. 239) and with radius equal to the actual length of an edge of the jiyramicl (that is, equal to 
v'd', Fig. 238), draw an indefinite arc, on which lay off the chords DC, C B, B A, A E, ED, equal 
respectively to the like -lettered edges of the base abcde; join the extremities of these chords with 
V: then -on Z» F lay off DP=d'p'; make C 0= EQ=^ d'o'" = the real length of c'o'; also BN= 
A M = d' m'" = the actual length of a'7n' and b'n'; join the jDoints P, 0, etc., thus obtaining the 
development of the outer boundary of the section. The laattern A^B^CDE^ of the base is obtained 
from the plan in Fig. 238, while NMq.^p.,o., is a duplicate of the shaded part of the sectional vieW' 
in the same figure. 

(7) In making a model of tire pyramid the student should use heavy Bristol board, and make 
allowance, wherever needed, of an extra width for overlap, slit as at x, y and z (Fig. 239). On thi.s 

I^ig-- 233. 




overlap put the mucilage which is to hold the model in shape. The faces will fold better if the 
Bristol board is cut half way through on the folding edge. 

397. For convenient reference the characteristic features of the Third Angle Method, all of which 
have now been fully illustrated, may thus be briefly summarized : 

(a) The various views of the object are so grouped that the plan or top view comes above the 
front elevation; that of the bottom below it; and analogously for the projections of the right and 
left sides. 

(b) Central, reference planes are taken through the various views, and, in each view, the distance 
of any point from the trace of the central plane of that view is obtained by direct transfer, with 
the dividers, of the distance between the same point and reference plane, as seen in some other 
view, usually the plan. 

398. To draio a truncated, pyramidal block, having a rectangular recess in its top; angle of sides, 
60°; lower base a rectangle 3" X 2", having its longer sides at 30° to the horizontal; total height 
•5%"; recess 1^" X ■^", and \" deep. (Fig. 240.) 

The small oblique projection on the right of the plan shows, pictorially, the figure to be drawn.. 



PROJECTION OF SOLIDS. — WORKIXG DRAWINGS. 



141 



The plan of the lower base will be the rectangle abde, 3" X 2", whose longer edges are inclined 
30° to the horizontal. 

Take AB and ma as the H- traces of auxiliary, vertical planes, perpendicular to the side and 
end faces of the Ijlock. Then the sloping face whose lower edge is d e, and which is inclined 60 ° 
to H, will have d,?/ for its trace on plane mn. A jjarallel to mn and ■^" from it will give -Sj, the 
auxiliary projection of the upj^er edge of the face sved, whence sv — at first indefinite in length — is 
derived, parallel to de. Similarly the end face btsd is obtained by projecting db upon AB at 6,, 
drawing b^z at 60° to AB and terminating it at Sj by CD, drawn at the same height (xVO "^^ 
before. A parallel to bd through s.^ intersects vs^ at s, giving one corner of the plan of the upper 
base, from which the rectangle stuv is completed, with sides parallel to those of the lower base. 




As the recess has ^-ertical sides we may draw its plan, o p q r, directly from the given dimen- 
sions, and show the depth by short -dash lines in each of the elevations. 

The orcUnary elevations are derived from the plan as in preceding problems; that is, for the 
front elevation, a'u's'd', by verticals through the plans, temiinating according to their height, either 
on a' d' or on lo's', ■^" above it. For the side elevation, e"v"t"b", with the heights as in the front 
elevation, the distances to the right or left of s" equal those of the plans of the same points from 
.5 i, regarding the latter as the h. t. of a central, vertical plane, parallel to V. 

The plane ST of right section, jjerpeyidicular to the axis KL, cuts the block in a section whose 
true .size is shown in the line -tinted figure g^h^kj^, and whose construction hardly needs detailed 
treatment after what has preceded. The shaded, longitudinal section, on central, vertical plane K L, 
also interprets itself by means of the lettering. 



142 



THEORETICAL AND PRACTICAL GRAPHICS. 



The true size of any face, as a u v e, raixy be shown by rabatment about a horizontal edge, as a e. 
As V is actually yV above the level of e, we see that v e (in space) is the hypothenuse of a triangle 
of base ve and altitude -^". Construct such a triangle, v^'^e, and with its h^ypothenuse v.^e as a 
radius, and e as a centre, obtain v ^ on a perpendicular to a e through v and representing the joath 
of rotation. Finding u^ similarly we have a^i^v^e as the actual size of the face in question. 

If more views were needed than are shown the student ought to have no difficulty in their 
construction, as no new principles would be involved. 

399. To draw a holloiv, pentagonal prism, 2" long; edges to be horizontal and inclined 35° to 
V; base, a regular pentagon of 1" sides; one face of the prism to be inclined 60° to H; distance 
lietween inner and outer faces, \". 




In Fig. 241 let H K h% parallel to the plans of the axis and edges; it will make 35° with a 
horizontal line. Perpendicular to HK draw mn as the h. t. of an auxiliary, vertical plane, upon 
which we may suppose the base of the prism projected. In end view all the faces of the prism 
would be seen as Ihm, and all the edges as points. Draw a,i6i, one inch long and at 60° to mn, 
to represent the face whose inclination is assigned. Completing the inner and outer pentagons, 
■allowing \" for the distance between faces, we have the end vie^^' complete. The plan is then 



PROJECTION OF S OLIDS. — W RKIN G DRAWINGS. 143 

obtained by drawing parallels to H K through all the vertices of the end view, and terminating all 
by vertical planes, a d and g h , 2)arallel to m n and 2 " apart. 

The elevations will be included between horizontal lines whose distance ajjart is the extreme height 
z of the end view ; and all points of the front elevation are on verticals through their plans, and at 
heights derived from the end view. The most expeditious method of working is to draw a horizontal 
reference line, like that of Fig. 243, which shall contain the lowest edge of each elevation; measuring 
upward from this line laj^ off, on some random, vertical line, the distance of each point of the end 
view from a line (as the parallel to ma through 6, in Fig. 241, or .c/y in Fig. 243) which repre- 
sents the intersection of the plane of the end view by a horizontal plane containing the lowest point 
or edge of the ol^ject; horizontal lines, tlirough the points of division thus ol^tained, will contain the 
projections of the corners of tlie front elevation, which may then be definitely located by vertical 
lines let fall from the plans of the same points. For example, e' and ./'', Fig. 241, are at a height, 
s, above the lowest line of the elevation, equal to the distance of e, from the dotted line through 
ij; or, referring to Fig. 243, which, owing to its greater complexity, has its construction given -more 
in detail, the distance upward from M to line G is ccjual to i/^y.. on the end view; from M to Q 
equals qiq,, and similarly for the -rest. 

Since the profile plane is omitted in Fig. 241 we take M' N' to represent the trace upon it of 
the auxiliary, central, vertical plane whose h. t. is M N; as already explained, all j^oints of the side 
elevation are then at the same level as in the front elevation, and at distances to the right or left 
of M' N' equal to the perpendicular distances of their plnm from M N. For example, e" s" equals e -s. 

The shade lines are located on the end view on the assumption that the observer is looking 
toward it in the direction HK. 

400. Projections of a hollow, pentagonal prism, cut by a vertical plane obli<pie to V. Letting the data 
for the prism be the same as in the last problem, we are to find what modification in the appear- 
ance of the elevations would result from cutting through the object by a vertical plane P Q (Fig. 242) 
and remoA-ing the part h x d i which lies in front of the plane of section. 

Each vertex of the section is on an edge of the elevation and is vertically 1)eliiw the point where 
PQ cuts the plan of the same edge; the student can,, therefore, readily convert the elevations of 
Fig. 241 into reproductions of those of Fig. 242 by drawing across the plan of Fig. 241 a trace P Q, 
similarly situated to the P Q of Fig. 242. . Supposing that done, refer in what follows to both 
Figures 241 and 242. 

Since PQ contains h we find h' as one corner of the section. Both end.t of tlie prism being- 
vertical, they will be cut by the vertical plane PQ in vertical lines; therefore h'l' is vertical until 
the top of the prism is reached, at I'. Join I' with x', the latter on the vertical througli .r — the 
intersection of PQ with the right-hand top edge ed, e'd'. From x the cut is vertical until the 
interior of the prism is reached, at o', on the line 5-4. AVe next reach w' on edge No. 4. The 
line o' w' has to be parallel to x'l' (two parallel planes are cut by a third in parallel line,s); but 
fi-om lo' the interior edge of the section is not parallel to I'h', since PQ is not cutting a vertical 
end, but the inclined, interior surface. Tlie other points hardly need detailed description, being 
similarly found. 

Tlie side elevation is obtained in accordance with the principle fully described in Art. 396 and 
summarized in Art. 397 (b). M' N' represents the same i)lane as 31 N; c"-s" ccjuals es, and anal- 
ogously f(ir otiier ])oints. 

401. Ill his elementary work in jn'ojections and sections of solids the student is recommended 
to lay an even tint of liurnt sienna, medium tone, over the projections of the ol)ject, after which 



144 



THEORETICAL AND PRACTICAL GRAPHICS. 



any section may be line -tinted; and, if he desires to further improve the appearance of the ^dews, 
distinctions may be made between the tones of the various surfaces by overlaying the burnt sienna 
with flat or graded washes of India ink. 




side elevation 
In' 



402. Projections of an L-slmped block, after being cut by a Tplane oblique to both V and H; the block 
also to be inclined to V and H, and to have nmning through it tiuo, non-communicating, rectangidar openings, 
whose directions are mutually perpendicular. 

If the dotted lines are taken into account the front elevation in Fig. 243 gives a clear idea of 
the shape of the original solid. The end view and plan give the dimensions. 

Requiring the horizontal edges of the block to be inchned 30° to V, draw the first line xy at 
60° to the horizontal; the pkns of all the horizontal edges will be perpendicular to xy. 

Let the inchnation of the bottom of the lilock to H be 20°. This is shown in the end view 
by drawing m,p, at 20° to xy. All the edges of the end vieAV of the object will then be parallel 
■or perpendicular to m^p^ and should be next drawn to the given dimensions. 



WOFKIXG DBAWIXGS. — PLAXE SECTIOXS OF SOLIDS. 



145 



The central opening, ?*,^/, /),",, through the larger part of the block, has its faces all J" from 
the outer faces. In the plan this is shown by drawing the lines lettered of at a distance of \" 
from the lioundary lines, which last are indicated as 1^" apart. 

The opening f/i'/'i-Si?, 
has three of its faces Y' 
from the outer surfaces 
of the block, while the 
fornlh, giJ'j, is in the 
same plane as the outer 
face /)\<?i. 

The cutting plane 
X Y gives a section 
which is seen in end 
view in the hnes c,f/i, 
(j,;, and Ic^l^; while in 
plan the section is pvo- 
jected in the shaded 
l^ortion, obtained, like 
all other parts of the 
p)lan, by perpendiculars 
to xy from all the 
points of the end view. 
For the front eleva- 
tion draw first the "reference line." To 
provide against overlapping of projections 
the reference line should be at a greater 
distance below the lowest point, /, of the 
plan, than the greatest height (a^a^) of 
the end view above x y. Then on MW 
lay off from 71/ the heights of the vari- 
ous horizontal edges of the l)lock, deriving 
them from the end ^n.ew. Thus OiO, is 
the height of A a' from M; from 31 to 
level B equals b^b.,, etc. Next project to 
the le^-el A fi-om points a a of tlie plan, 
getting edge a'o' of the elevation, and 
similarly for all the other corners of the 
block. Xofice that all lines that are parallel 
Oil the object icill be parallel in each projec- 
tion (except when their projections coin- 
cide) ; also that in the case of sections, those 
onflines iciH be parallel irhich are the inter- 
section of pen-all cl jjlanes by a third plane. 
These principles may he ad\-antageously employed as checks on the accuracy of the construction by 
points. The construction of the side elevation is left to the student. 




146 



THEORETICAL AND PRACTICAL GRAPHICS. 



^'ig-. 2.-^^. 




FRONT ELEVATION. 

"With section made by vertical plane P Q 

££ Reference line p- 



SIDE ELEVATION. 

With section by plane S T. Shade 
lines on this view are located for ^ 
pictorial effect and not in accord- 
ance with shop rule. 



WORKING DRAWINGS. — PLANE SECTIONS OF SOLIDS. 147 

40^:5. Projeriiims and .^sections of a block (if irrcf/tilar form, with two nwtiiaUij perjjendicular openings 
through it, and loith equal, square frames projecting from each .i-ide. 

In Fig. 244 the .side ele^-ation shows clearly the object dealt witli, while we look to the end 
view for most of the dimensions. The large central opening extends from w^iv^ to x^y^. The width of 
the main portion of the block is shown in plan as 2^", Ijetween the lines lettered ae. The square 
frames project ^" from the sides, while the width of the central opening between the lines wa; is f". 

Two section planes are indicated, ST across the end view, and PQ — a vertical jjlane — across the 
plan : the section made l\v plane S T is, however, shown only in fringed outline on the plan, though 
fully represented on the side elevation. The front elevation shows the section made by j)lane P Q, 
with the visilile portion of that part of the oljject that is behind the cutting plane. 

Although detailed explanation of this problem is unnecessary after what has preceded, yet a 
brief recapitulation of the various steps in the construction of the views may l^e appreciated by some, 
Ijefore passing on to a more advanced topic. 

(a) E F, the first line to draw, is the trace of the vertical plane on which the end view is 
projected, and is at an angle of 60 ° to a horizontal line in order that the edges of the object (as 

a a, hb e e) may be inclined at 30° to the front vertical plane, which we may assume as one 

of the conditions of the problem. 

(b) A rotation of the oliject through an angle 0° about a horizontal axis that is perpendicular 
1o E F, as, for examj-le, the edge through f, is shown Ijy the inclination of tlie end ^iew to EF 
at an angle a , _f , E = 6°. 

(c) Drawing the end view at the required angle to E F we next derive the plan therefrom by 
perpendiculars to E F, tenninating them on jjarallels to E F (as the lines a e, wx, nh, etc.,) whose 
-distances ajjart conform to given data. 

(d) The elevations. For these a common reference line E' F'f" is taken, horizontal, and sufficienth^ 
bielow the plan to avoid an overlapping of \'iews. 

For the front elerntion any point, as '/, is found vertically below its lAan b, and is as far from 
E' F' as 6, is — perpendicularly — from E F. 

The height at which the section j^lane P Q cuts any line is similarly obtained. Thus at z it 
•cuts the vertical end face of the block in a line which is carried over on the end view in the 
indefinite line Zz.,; the portions of Zz.^ which lie on the end view of the frame ffihii^ are the 
•only real jiarts to transfer to the front elevation, and are seen on the latter, vertically below z and 
running from z' down; their distances from E' F' being simply those from Z, on the end view, 
transferred. 

The .side elevation. Any point or edge is at the same Icrel on the side elevation as on the 
front; hence the edge through b" is on b' b' produced. The distances to the right or left of M' N' 
equal those of the corresponding points on the plan from M N ; thus o"j' equals oj, etc. 

404. Changed planes of projections. In the problems of Arts. 399-403 the employment of an 
"end ^-iew" — which was simply an auxiliary elevation — has prepared the student for the further use 
of ijlanes other than the usual planes of iDrojection; and if the au:ciliary ptlan is now mastered he 
is prepared to deal v.-\X\\ any case of rotation of object aliout \-ertical or horizontal axes, since new 
■and ]iro])erly located planes of i^rojection are their practical equivalent. 

In Fig. 245 the object is represented in its initial position by the line-tinted figures marked 
" first plan '' and " first elevation." The third and fourth elevations show somewhat more pictoriaUy 
that it is a Imlldw. truncated, triangular prism, having through it a rectangular opening that is per- 
pendicular to the front and rear faces. 



148 



THEORETICAL AND PRACTICAL GRAPHICS. 




CHAXGIXG PROJECTION-PLANE EQUIVALENT TO ROTATING OBJECT. 149 

(a). Rotation about a vertical axis, or its equivalent, a change in the vertical plane of 'projection. 
Result : second elevation derived from first plan and elevation. 

Let the axis be one of the vertical edges of the object, as that at d in the first plan ; also let 
the rotation be through an angle Ydo or 0°, {6 being taken, for convenience, equal to the angle 
YaF, which — with the line p g — will be employed in a later construction). If we were actually to 
rotate the object through an angle 6 the new pAan would be the exact counterpart of the first, but 
its horizontal edges would make an angle with their former direction, and the new elevation 
would partly overlap the first one. To avoid the latter unnecessary complication, as also the 
duplication of the plan, we make the first plan do double duty, since we can accomplish the 
equivalent of rotation of the oliject by taking a new vertical plane that makes an angle with the 
plane on which the first elevation was made. This equivalence will be more evident, if some small 
object, as a piece of india rubber, is placed on the " first plan " with its longer edges parallel to 
a b, and is then viewed in the direction of arrow No. '2 through a pane of glass standing vertically 
on XZ; after which turn both the object and the glass through the angle 6 until the glass stands 
vertically on e'j' and then view in the direction of arrow No. 1. 

The second plane ma}' be located anywhere, as long as the angle 6 is preserved; XZ, making 
angle 6 at x^ with e'j', is, therefore, a random position of the new plane, and the projection upon 
it is our "second elevation." 

Since the heights of the various corners of an object remain unchanged during rotation about a 
vertical axis we will find all points of the second elevation at distances from the reference line XZ 
that are derived from the first elevation, and laid off on lines drawn perpendicular to XZ from the 
vertices of the j^lan: thus aO is perpendicular to XZ, and a" equals o'a'; c"J'=c'j', etc. 

(b). Rotation about a horizontal axis, or its equivalent, the adoption of a new horizontal plane. 
Result: second pilan derived from first plan and second elevation. 

Having in the last case illustrated the method of complying with the condition that rotation 
should occur through a given angle (which is incidental!}' shown again, however, in the next con- 
struction) we now choose an axis p ci so as to illustrate a diiferent kind of reijuirement, \iz.: that 
during rotation the heights of any two points of the object, which were at first at the same level as 
the axis, shall be in some laredetermined ratio, regardless of the amount of rotation. In the figure 
it is assumed that e'(d) is to l:ie at one -fifth the height of j'j, and that rotation shall occur aliout 
an axis passing through the lower end o' of the vertical edge at a. By drawing ad and aj, 
dividing the latter into five equal parts, and joining d with n — the first point of division from a — 
we obtain the direction dn, j^arallel to A\-hich the axis pq is drawn through a. The distance dp is 
then one - fifth of ,/' q , and they shorten in the same ratio, as rotation occurs. 

After locating the axis the next step is, in^-ariably, the drawing of an elevation upon a plane 
perpendicular to the axis. This we happen, however, to have already in our " second elevation," 
having, in the interest of compactness, so taken 6 in the preceding case that the vertical plane XZ 
would l)e peri:)endicular to the axis we are now ready to use. 

Any rotation of the object alDOut pq will, evidently, not change the fortn of the "second 
elevation" Init simply incline it to X Z. But, as before, instead of actually rotating the object, 
which would prolnaljly give projections overlapjoing those from which we are working, we adojit a 
new plane MN as a horizontal jjlane of projection, so taken that it fulfills either of the following 
conditions : (a) tliat the oliject should l^e rotated about p q through an angle /3 ; (1)) that the 
corner J' should lie higher than by an amount x, MN being drawn tangent to an arc having 
J' for its centre, and J'f (equal to x) for its radius. 



150 



THEORETICAL AND PRACTICAL GRAPHICS. 



:Fig-. a-is. 



-.p 



Reference to Fig. 246 ma_y make it clearer to some that MN is the trace of the new plane 
upon the vertical plane whose h. t. is XZ; that N lies vertically below the line XZ and is as truly 
perpendicular to the axis of rotation as is XZ; also that in Fig. 245 a view in the direction of 
arrow No. 3 (i. e., perpendicular to MN') is equivalent to a view 
perpendicular to the plane V in Fig. 246 after the whole assem- 
blage of planes and object has been rotated together about 
HOH until the "new jDlane " takes the position out of 
which the first horizontal plane has just been rotated. 

(The remainder of the references are to Fig. 245.) 

The second plan is obtained by drawing P^Qi, 
parallel to MN, to represent the transferred 
trace P Q of a vertical reference plane taken 
through some edge h and parallel to the 
plane Z N of the second elevation ; 
then any point d^ is as far from 
P, Q, as the same point d, on 
tlie first plan is from P Q ; 
that is, rfjO, equals od, and 
similarly for other points. 

(c) Further rotation about ver- 
tical axes. To show how the foregoing processes ma}^ 
be duplicated to any desired extent let us suppose 
that the object, as represented by the second plan 
and elevation, is to be rotated through an angle <^ 
about a vertical line through h,. If the rotation 
actually occurred, the jDlan b^G^ would take the posi- 
tion 6i(?2, and the other lines of the plan would take corresponding jjositions in relation to a vertical 
plane on P, Qj. A new vertical plane on b^Q,, at an angle <t> to b^G^, will, however, evident^ 
hold the same relation to the plan as it stands, and transferring such new jjlane forward to O'Pj 
we then obtain the points of the new (third) ele^'ation by letting fall perpendiculars to 0' R^ from 
the vertices of the second plan, and on them laying off heights abo-\'e O'P, equal to those of the 
same points above MN in the second elevation. Thus j'9 equals /'/; TF'6 in the fourth equals 
IF" 6 in the second. 

The fourth elevation is a view in the direction of arrow No. 5, giving the equivalent of a ninety - 
degree rotation of the object from its last position. To obtain it take a reference line r r through 
some point of the second plan, and parallel to O'R,; then R R' rejDresents the vertical jjlane on r r, 
transferred. From R R' lay off — on the levels of the same points in the third elevation — distance 
1 C" = c.X\; 4 TF" = H-^jT'F], as in j^receding analogous constructions. 




THE DEVELOPMENT OF SURFACES. 

405. The developjment of surfaces is a topic not altogether new to the student who has read 
Chapter V and the earlier articles of this chapter;* so far, however, it has occurred only incidentally, 
but its importance necessitates the following more formal treatment, which naturally precedes problems 
on the interpenetration of surfaces, of which a develoi^ment is usually the jDractical outcome. 



*The following articles should be carefully reviewed at this point: 120; 191; 314-6; 3S9, and Case 6 of Art. 3%. 



THE DEVELOPMENT OF SURFACES. 



151 



A development of a surface, using the term in a practical sense, is a i^iece of cardboard or, more 
generally, of sheet -metal, of such shape that it can be either directly rolled up or folded into a 
model of the surface. Mathematically, it would be the contact- area, were the surface rolled out or 
unfolded upon a plane. 

The "shop" terms for a developed surface are "surface mi the flat" "stretch-out," "roll -out"; 
also, among sheet -metal workers it is called a pattern; but as pattern -making is so generallj^ under- 
stood to relate to the patterns for castings in a foundry, it is best to emijloy the cjualifying words 
sheet-metal when desiring to avoid any possible ambiguity. 

406. The mathematical nature of the surfaces that are capable of devel(5pment has Ijcen already 
discussed in Arts. 344-346. Those most frec^uently occurring in engineering and architectural work 
are the right and ol)lique forms of the pyramid, prism, cone and cylinder. 

407. In Art. 120 the development of a right cylinder is shown to be a rectangle of base equal to 
'2-n-r and altitude A, where h is the height of the cylinder and r is the radius of its base. 

408. The development of a right cone is proved, by Art. 191, to be a circtdar sector, of radius 
equal to the slant height R of the cone, and whose angle & is found by means of the projjortion 
R : r :: 360° : 0; r being the radius of the base of the cone. 

409. The development of a right pyramid is illustrated in Art. 389 and in Case 6 of Art. 396. 

410. ^^'e next take up right and oblique prisms, and the oblique pyramid, cone and cylinder; 
while for the sake of completeness, and departing in some degree from what was the plan of this 
work when Art. 345 jjassed through the i^ress, the regular solids will receive further treatment, and 
also the developable helicoid. 

411. The development of a right prism. Fig. 
247, represents a regular, hexagonal prism. The 
six faces being equal, and e b c f showing their 
actual size, we make the rectangles A BCD, 
Bi BEFC, etc., each equal to ebcf; then AA^ 
f '^ equals the perimeter of the upper l)ase, and we 

have the rectangle A.A.^B^D for the development sought. 

412. The deveJop/ment of a right prism, beloiv a cutting plane. 
last article develop first as if there were no 
section to be taken into account. This gives, as 
before, a rectangle of length A A, and of altitude 
a d, divided into six equal jiarts. Then project, 
from each point where the jilane cuts an edge, 
to the same edge as seen on the development. 

413. Right Section. Rectified Curve. Developed Curve. A plane perpendicular to the axis of a. 
surface cuts the latter in a right section. The bases of right cones, pyramids, cylinders and p)risms 
fulfil this condition and require no special construction for their determination; but the development 
of an o):ili(jue i'orm usually involves the construction of a right section and tlien the laying off on 
a straight line of a length equal to the jjerimeter of such section. Should the right section Ije a 
curve its equivalent length on a straight line is called its rectification, which should not be confounded 
with its develojmient, the latter not Ijeing necessarily straight. 

414. The development of an oblique pjrisni, when the faces are equal in tvidth. In Fig. 249, an 
oblicjue, hexagonal prism is shown, with x for the width of its faces. Since the perimeter of a 
right section would evidently equal 6 x we may directly lay off x six times on some iierfjendicular 





.,=!„„- 




a A 


B 


E H 




t 




s»== 


~S! 


d 
















^ 


t'l 




I 


3 




C 




F C 


3 







Ai 



Taking the same prism as in the 




a f 



152 



THEORETICAL AND PRACTICAL GRAPHICS. 



:Fig-. 3-^9. 




E'ig-- SSO. 



to the edges, as that through a. The seven i^arallels to a b , drawn at distances x apart, will contain 
the various edges of the prism as it is rolled out on the plane; and the position of the extremities 
are found by perpendiculars from their original positions. 
The initial position o j 6 , is parallel to but at anj^ dis- 
tance fi'om a b. 

415. The development of an oblique prism whose faces 
are unequal in ividth. 

In Fig. 250 c' d' h' g' is the elevation of the prism; 
n p a plane of right section. To get the true shajDe, 
1-2-3-4, of the right section, we require abhfe — 
l^lan of the i:)rism.' Assuming that to have been given, 
imagine next a vertical reference - plan standing on ab. 
The right section plane np cuts the edge c' d' at n, 
which is at a distance x in front of the assumed reference - plane. Make n 2 = x. Similarly make 

oo = y, and p4 = 2; then 1-2-3-4 is the right section, seen 
in its true size after being revolved about the trace of the 
right-section plane upon the assumed reference -plane. 

Prolong jj n indefinitely, and on its extension make 
l'-2'=l-2; 2' 3' = 2-3, etc. Parallels to c' d' through 
the points of division thus obtained will contain the 
edges of the developed larism, and their lengths 
are definitely determined by perijendiculars, as 
■"-,^ h'h", ff", from the extremities of the orig- 

inal edges. 

416. The development of an oblique 
cylinder. Let am'n'k, Fig. 251, be an 
oblique cylinder with circular base. Take 
any plane of right section, as a'k'. 
Draw various elements, as those through 
b', c', etc., and from their lower extrem- 
ities erect perpendiculars to ak, as cc,, 
terminating them on the arc af^k, which 
represents the half base of the cylinder. 
On c c' make c'c" = cc^; on e e' take 
e' e" = ee^, and similarly obtain other 
points on the elements, through which 
the curve a' c" e" y" I' can be drawn, this being one -half of the curve of right section, shown after 
revolution about its shorter diameter. Making KA equal to the rectified semi -ellipse just obtained, 
lay off A B = am ah^; i? C = arc b-^Ci, etc., and through the points of division thus obtained on KA 
draw indefinite parallels to the axis of the cjdinder. These will represent the elements on the 
development, and are limited by the dotted lines drawn perpendicular to the original elements and 
through their extremities. 

The area a.^k.,NM is the development of one -half of the cylinder, the shaded area representing 
all between a' h' and the base al\ 




* In the iuterest of compactness the "Fii-st Angle" position of the views is ;employe(l in Figs. '250, 253 and 



THE DEVELOPMEST OF SURFACES. 



153 



417. The development of an oblique pyramkl. 




The development will evidently consist of a series of 
triangles having a common vertex. To ascertain 
tlie length of any edge we may carry it into or 
parallel to a plane of projection. Thus in Fig. 2.52 the 
edge vb is carried into the vertical jjlane at vh". Its 
true length is evidently the hypothenuse of a right- 
angled triangle of base o /j = n J ", and altitude v o . 
In Fig. 253 a pj^ramid is shown in plan and 
elevation. Making 



V c to 
length 



a" = v a we have 
(■'((" for the actual 
lengtli of edge v'a', a 
construction in strict 
>'^ analogy to that of 
Fig. 252. The ])lan 
V b being jjarallel to 
the base line shows 
that v'b' is the 
actual length of that 
edge. By carrying 
vci, where it becomes parallel to V, and then projecting Cj to 
of edge v'c'. x'lg-- sss. 



E"ig-. 2S2. 




?" we get v'c" for the true 




To illustrate another method make vv^ = v'o; then v^d 
bv rabatment into H. 



is the real length of r' <l', shown 



154 



THEORETICAL AND PRACTICAL GRAPHICS. 





For the development take some point v, and from. it as a centre draw arcs having for radii the 
ascertained lengths of the edges. Letting v.^A represent the initial edge of the development take A 
as a centre, ad as a radius, and cut the arc of radius v^d at D ; then Av,D is the development of 
the face avd, a'v'd'. With centre D and radius dc obtain C on the arc of radius v' c" ; similarly 
for the remaining faces, completing the development v^ — AD...A^. 

The shaded area v^ — TP...T is the development of that i^art of the pyramid above the oblique 
plane s'jj', found by laying off, on the various edges as seen in the develoisment, the distances along 
those edges from the vertex to the cutting plane; thus v.^S = v' s'; v,_N = v' n', the real length of 
v' m' ; »2-f='''iPi) fhe length of v' p\ etc. 

418. The development of an oblique cone. The usual method of sohing this problem gives a result 
which, although not mathematically exact, is a sufficiently close 
approximation for all practical jjuriDOses. In it the cone is treated 
as if it were a i^yramid of many sides. The length of any 
element is then found as in the last problem. Thus in Fig. 264 
-an element v c is carried to v c" about the vertical axis v o. 

In Fig. 265 we have v' a g for the elevation of the cone, and 

v — abc... g for the half 

plan. Make ob" = 

ob; then v' h" is 

the real length 

of the element whose plan is o b. Similarly, 
c, d, e and f are carried by arcs to ag and 
there joined with v'. 

For the development make v^A 
equal and parallel to v' a, and at 
any distance from it. With v^ as 
a centre draw arcs with radii equal 
to the true lengths of the elements; 
then, as in the pyramid, make 
A B = arc ab; B C = arc be, etc. 
The greater the number of divisions on the semi -circle ab...g the more closely will the develop- 
ment ai^i^roximate to theoretical exactness. s'lg-- sse. 

419. The five regular convex solUU, with the forms of their 
developments, are illustrated in Figs. 266-265. They have 
already been defined in Art. 345, and that five is their limit 
as to number is thus shown: The faces are to be equal, 
regular polygons, and the sum of the jjlane angles forming a 
■solid angle must be less than four right angles; now as the 
angles of equilateral triangles are 60° we may evidently have 

grou]DS of three, four or five and not exceed the limit; with dodecahedron. 

squares there can be groujjs of three only, each 90°; with 
regular pentagons, their interior angles being 108°, groups of 
three ; while hexagons are evidently impracticable, since three of 
their interior angles would exactly equal four right angles, adapt- 
ing them perfectly — and only — to plane surfaces. (See Fig. 131.) 



^'ig-- SST. 




X'ig- SSS. 



TETnAHEDRON. 






^^igr- 2SO_ 




OCTAHEDRON. 



ICOSAHEDRON. 



THE FIVE REGULAR CONVEX SOLIDS. 



155 



The dihedral angles between the adjacent faces of regular solids are as follows: 70° 31' 4-1:" for 
the tetrahedron; 90° for the cube; 109° 28' 16" for the octahedron; 116° 35' 




TETRAHEDRON. 

For the tctnihednm 
" " oHuhedivii 
of an inscrilied cube 

E-ig-- 233. 



54" for the dodecahedron; and 138° 11' 23" for the icosahedron. 

A sjohere can be inscribed in each regular solid and can also 
as readily be circumscribed about it. 

The relation between d, the diameter of a sphere, and e, the 
edge of an inscril>ed regular solid, is illustrated graphically by 
Fig. 266, l.)ut may l3e otherwise expressed as follows: 



x'ig-. ass. 




d 



: e 
d : c 
when 



V 


M\ 


_-_ 


A^ 




\ 


"m 




/ w 


\, 








■,/\ 


J\ 




OCTAHEDRON. 



DODECAHEDRON 



:: n/3 : ^/ 2 ; for the cube d : e :: >/ 3 : 1 

: : •>/ 2 : 1 ; " " dodecahedron e = the greater segment of the edge 

the latter has been medially divided, that is, in" extreme and mean ratio. 
For the icosahedron e = the 
chord of the arc whose tangent is 
d; i. e., the chord of 63° 26' 6". 
Reference to Figs. 256-260 and 
the use of a set of cardboard 
models which can readily be made 
bj' means of Figs. 261-265 will 
enable the student to verify the 
following statements as to those ordinary ^-iews whose construction ^^■n\\\d naturally precede the 
solution of problems relating to these surfaces. 

E-ig-. SS5. In all liut the tetrahedron each 

face has an ecjual, opposite, parallel 
face, and excej^t in the cube such 
faces have their angular points alter- 
nating. (See Figs. 260, 267, 268.) 
The tetrahedron projects as in 
Fig. 256, upon a plane that is par- 
allel to either face. 
The ciibe projects in a square upon a jjlane parallel to a face, 
while on a plane perjoendicular to a body diagonal it projects as 
a regular hexagon, with lines joining three alternate yei-tices with the centre. 

The octahedron, which is practicallj' two equal square jDyramids with a common base, projects in 
a square and its diagonals, ujjon a plane perpendicular to either body diagonal; in a rhombus 
and shorter diagonal when the plane is parallel to one body diagonal and at 45° with the other 

^S-S- 2ST- X^ig-. 2SS- X'ig'. 3S3. 




ICOSAHEDRON 







two; and (as in Fig. 267) in a regular hexagon with inscribed triangles (one dotted), when it is 
projected ujjon a plane parallel to a face. 

The dodecahedron projects as in Fig. 268 whenever the plane of projectioii is j^arallel to a face. 



156 



THEORETICAL AND PRACTICAL GRAPHICS. 



IFigr- STO. 




Fig. 260 represents the icosahedron i^rojected on a jjlane parallel to a face, and Fig. 269 when the 

projection -plane is perpendicular to an axis. 

420. The Devdopahle Hclicoid. When the word heUcoid is used without qualification it is under- 
stood to indicate one of the ivarped helicoids, such as is met with, for example, in screws, spiral 

staircases and screw propellers. There is, however, 
a developable helicoid, and to avoid confusing it with 
the others its characteristic property is always found 
in its name. As stated in Art. 346 it is generated 
1:),V moving a straight line tangentially on the ordi- 
nary helix, which cur^-e (Art. 120) cuts all the 
elements of a right cylinder a;t the same angle. 
Fig. 209 illustrates the completed surface pictorially ; 
Fig. 270 shows one orthographic jjrojection; and in 
Fig. 271 it is seen in process of generation by the 

hypothenuse of a right-angled triangle that rolls tangentially on a cylinder. 

The construction just mentioned is based on the property of non- plane curves that at any point 

the curve and its tangent make the same angle with a given plane; if, therefore, the helix, beginning 

at o, crosses each element of the cylinder at an 

angle equal to ob p in the rolling triangle, the 

hypothenuse of the latter will evidently vaoxe not 

only tangent to the cylinder, l^ut also to the helix. 
The following important properties are also 

illustrated by Fig. 271 : 

(a) The involute* of a helix and of its hori- 
zontal projection are identical, since the point !i is 
the extremity of both the rolling lines, o b and p.b. 

(b) The length of any tangent, as 'm 6, is that 
of the helical arc m a on which it has rolled. 

(c) The horizontal projection b q of any tan- 
gent b m equals the rectification of an arc a q which 

is the projection of the helical arc from the initial point a to the point of tangency to. 

The development of one nappe of a helicoid is shown in Fig. 273. It is merely the area between 
a circle and its in\-olute; but the radius p, of the base circle, equals r sec'- ^,t in which r is 




* For full treatment of the involute of a circle refer to Arts. 186 and 187. 

t This relation is due to considerations of curvature. At any point of any curve its curvature is its rate of 
departure froni its tangent at that point. Its radius 0/ curvature is that of the osculatory circle at that point. (Art. 
380.) Now from the nature of the two uniform motions imposed upon a point that generates a helix (Art. 120) 
the curvatui-e of the latter must be uniform; and if developed upon a plane by vieans of its curvature it must 
become a circle — the oiily plane curve of uniform curvature. The radius of the developed helix will, obviously, be 
the radius of curvature of the space helix. Following Warren's method of proof in establishing its value let 
a, b and c (Fig. 272) be three equi-distant points on a helix, with & on the foremost element; then a' c' is the 
■elevation of the circle containing these points. One diameter of the circle a' b' c' is projected at b'. It is the 
hypothenuse of a right-angled triangle having the chord 6 c, 6'c', for its base. Let 2 p be the diameter of the 
circle a'b'e'; 1r = bd, that of the cylinder. Using capitals for points in space we have B~C' =2pXbe; also iTc^ = 
2rXbe; whence, dividing like members and substituting trigonometric functions (see note p. 31), we have 
p = rsec = /3, in which p is the angle between the line £C and its projection. 

Let 9 be the inclination of the tangent to the helix at b'. If, now, both A and C approach B, the angle 
3 will approach as its limit; and when A, B and C become consecutive points we will have p = r3ec=e = the 
radius of the osculatory circle = the radius of curvature. 

For another proof, involving the radius of curvature of an ellipse, see Olivier, Cours de Geometric Descriptive, 
Third Ed., p. 197. 



T^ig-. 




THE INTERSECTION OF SURFACES. 



157 



^ig-. ST-i- 



m 


n 


/ 


/ 




% 



the radius of the cyUnder on which the heUx originall}' lay, and 6 is the angle at which the 

x-ig- avs. helix crosses the elements. To de- 

termine p draw on an elevation of 
the cylinder, as in Fig. 274, a line 
a h, tangent to the helix at its fore- 
most point, as in that position its 
inclination 6 is seen 
in actual size ; then 
from 0, where a b 
crosses the extreme 
element, draw an in- 
definite line, OS, par- 
allel to c d, and cut it at m by a 
line a m that is perpendicular to a 6 
at its intersection with the front 
element cf of the cylinder; then 
m = p = r sec.- 6. For we haA-e 
oa = 7) sec 9 = r sec ; and o a (= )•) : oa :: o a : o m ; whence o m = r sec ' 6 = p. 

The circumference of circle p equals 2 ir r sec 0, the actual length of the helix, as may he seen 
by developing the cylinder on which the latter lies. The elements which were tangent to the hehx 
maintain the same relation to the developed hehx, and appear in their tme length on the development. 
The student can make a model of one nappe of this surface by wrapping a sheet of Bristol 
board, shaped like Fig. 273, upon a. cylinder of radius r in the equation r sec' e = p; or a two- 
napped hehcoid by superposing two equal circular rings of paper, binding them on their inner edges 
Tvith gummed paper, making one radial cut through both rings, and then twisting the inner edge 
into a helix. 




THE INTERSECTION OF SURFACES. 

421. When plane-sided surfaces intersect, their outhne of interpenetration is necessarily composed of 
straight hnes; but these not being, in general, in one plane, form what is called a tioisted or toarped 
polygon; also called a gauche polygon. 

422. If either of two intersecting surfaces is curved their common line will also be curved, 
except under special conditions. 

423. AMien one of the surfaces is of uniform cross section — as a cylinder or a prism — its end 
\iew will show whether the surfaces intersect in a continuous line or in two separate ones. In Cases 
a, b, c, d and (/ of Fig. 275, ^\heve the end ^-iew of one surface either cuts but one limiting line of 
the other surface or is tangent to one or both of the outhnes, the intersection will be a continuous 
line. Two separate curves of intersection will occur in the other possible cases, illustrated by e and 

/, in which the end view of one surface either crosses both the outlines of the other or else hes 
wholly lietween them. 

A cylinder will intersect a cone or another 
oj^linder in a plane curve if its end view is tangent 

to the outlines of the other surface, as in d and x-ig.. st-s. 

^, Fig. 275. Two cones may also intersect in a plane curve, but as the conditions to be met are 
not as readilv illustrated they will l)e treated in a special prolilem. 




158 



THEORETICAL AND PRACTICAL GRAPHICS, 



■Fi-s- SVS- 



Referenee Line 



424. In general, the line of intersection of two surfaces is obtained, as stated in Art. 379, by 
passing one or more auxiliary surfaces, usually planes, in such manner as to cut some easily con- 
structed sections— as straight lines or circles— frOm each of the given surfaces; the meeting - points of 
the sections lying in any auxiliary surface will lie on the line sought. 

The application of the principle just stated is much simplified whenever any face of one or 
other of the surfaces is so situated that it is projected in a line.' This case is amply illustrated in 
the problems most immediately following. 

The beginner will save much time if 
he will letter each projection of a point as 
soon as it is determined. 

425. The intersection of a vertical triangular 
prism by a horizontal square prism; also the 
developments. 

The vertical prism to be 1|" high and 
to have one face parallel to V; bases equi- 
lateral triangles of 1" side. 

The horizontal prism, to be 2" long, its 
basal edges f", and its faces inclined 45° to 
H; its rear edges to be parallel to and 
i" from the rear face of the horizontal 
prism. 

The elevations of the axes to bisect 
each other. 

Draw e i horizontal and 1" long_ for the 
plan of the rear face of the vertica} prism. 
Complete the triangle egi and project to 
levels 1^" apart, obtaining e'f, g'h',-i'j', 
on the elevation. 

Construct a^r^end view g" i" j" h", using 
t" j" to represent the reference line et, 
transferred. 

The end view of the horizontal prism 
is the square a" b" c" d", having its diagonal horizontal and 
upper and lower bases of the other prism, and with its corner 
from i" j". The plan and fi'ont elevation of the horizontal 
rived from the end view as in preceding constructions. 

Since the lines eg and gi are the plans of vertical faces 
their intersection by the edges a, b, e and d of the hori- 
n, m, I, p, q, r — aird project to the elevations of the same 
edge a a meets the other prism at o and h, Avhich project 
o' and h'. Similarly for the remaining points. 

The development of the vertical prism is shown in the shaded rectangle IJ, of length Sgri and 
altitude e'f. (See Art. 411). The openings 0, j^i g, r, [and h^l^m^n^ are thus found: For p^, which 
represents p', make Ppi = xp', and G P = g p, the true distance of p' from g' h' ; similarly, qiO = 
q' y, and G = o g. 





!>= midway between the 

b " one - eighth inch 

"= prism are next de- 

'^- of one prism we note 
zontal prism — as at 

"' edges. Thus the 
to the level of a" at 



THE IXTEBSECTIOX OF SURFACES. 



159 




The right hidf uf the horizontal iirism, u' a' c' q', is developed at r .,h .^b ./r ^ after the method of 
Art. 412. 

42(3. The iiitcrscrthiii uf tiru jiri><in--<, mic vertical, the other horizontal, each having an edge extericrr to the 

■other. The condition just made will, as already stated (Art. 423), make the result a single warped polygon. 

E^g-- S.77. Let a bed. 1" x ^", he the plan of the vertical prism, which 

_«/-A"_« stands with its broader faces at some convenient angle c^u g to V. 

From it construct the front and side elevations, taking a reference 
plane through d for the latter. 

Let the horizontal prism be triangular (isosceles section) one 
face inclined 45° to H; another 30° to H; the rear edge to be 
i" from tliat of tlie vertical prism. 

Begin Ijy locating g" one -fifth inch from the right edge, draw 

; I f"g" at 45°, making it of sufficient length to have /" exterior to 

j j j the other prism; then f'e" at 30° to H, 

Jsll [ L c b^ c"a" d" terminated at e" by an arc of centre g" 

and radius g"f"; finally e" g". The edges c', 
/' and g' of the front elevation are then 
l^rojected fi-om e", /" and g". 

The rear edge g in the plan meets the 
face fl d at s, which projects to -s' on the 
edge g'. 

Jlo\-ing fijrward from s, the next edge 
reached, of either solid, is <i, of the vertical 
prism. To ascertain the height at which 
it meets the Other prism we look to the 
end -iiew, finding q" for the entrance and 
t" for the exit. Being on the way up fi-om g" to c" we use q" , reser\ing t" until we deal with 
the face g" f". Projecting q" over to q' draw r/.s', dotted, since it is on a rear face. 

Returning to a and mo\-ing toward b we next reach the edge e, whose intersection p witli a b 
is then projected to edge e' at p' and joined with q'. 

For the next edge, h, we 
obtain o' from the side eleva- 
tion, projecting from the inter- 
section of /"(?" by the edge h". 
Moving from Ji toward ic, 
projecting to the front ele^"a- 
tion from either the plan or 
the side elevation according as 
we are dealing with a hori- 
zontal edge or a \-ertical one, 
we complete the intersection. 

The develojiment of the 
vertical prism is shown in Fig 
fully described, r?f?, = perimeter abed; a q= n' Q' ; h =^ b' o' ; a. 
distance of -s' fi-f)m o'c', etc. 





•p' 


__u__ 






«■' 




e" 


1 

1 

1 

1 

1 

1 

1 

1 
, 1 
\i 
\i 

V" 




1 '' ' 








^~^''--/' 




/'' 


"/^ 


30° \ ^ 


V 1 


. w 


/ 1 1 


/ 
, / 
1 / 
' / 


\ 






t' 


-Vr- 


> 




il' 






f 








i 



■s".s- s7-e- 

d X a y b z c d 














1 / ■ 


1 .'^ ■ ^ 


u 


"^-YVV 



s-xgr- ST"©. 




278. As akeady 
a X (of Fig. 277) ; a; 5' = \-ertical 



160 



THEORETICAL AND PRACTICAL GRAPHICS. 



Although not required in shop work the draughtsman will find it an interesting and valuable 
x-ig-. seo. exercise to draw and shade either soHd after the removal of the other ; also to 

draw the common [solid. The former is illustrated by Fig. 279; the latter 
l)y Fig. 280. 

427. The intersection of two prisms, one vertical, the other oblique but with 
edges parallel to V. 

Let abcd....a'r' (Fig. 281) be the plan and elevation of the vertical prism. 
Let the oblique prism be inchned 30° to H; its faces inclined 60° and 30° 
respectively to V; its base a rectangle 1^" X f", and its rear edge yV back of 
the axis of the vertical prism. 

Through some point o' of the edge e' o' draw an indefinite line, o' f, at 30° to H, for the 
elevation of the rear edge, and //, also indefinite in length at first but ^" back of s, for the plan. 




^igf. 2S1. 





,,'-'' Take a reference plane MN through 

s and as in Art. 397 (b) construct an 
auxiliary elevation on 71/ iV, transferring it so 
that it is seen as a perpendicular to o'/', thus obtaining 
the same view of the prisms as would be had if looking 
in the direction of the arrow. To construct this make o" f" 
equal to A") draw f'i" at 60° to MN and on it com- 
plete a rectangle of the given dimensions; after which lay 
off the points of the pentagonal prism at the same distances 
from MN in both figures. Project back, in the direction 
of the arrow, from /", g", h" and i" to the front elevation, 
and draw g' i' and the opposite base at equal distances each side of o'. 

For the intersection we get any point n' on an oblique edge, as g', by noting and projecting from 



THE IXTERSECTIOX OF PLAXE-SIDED SURFACES. 



161 



E-ig-. 2Sa. 




E'ig-- 2S3. 
X 



71 where the plan gg meets the foce c d. For a vertical edge as c' m' look to the auxiliary 
elevation of the same edge, as c", getting /" and m" which then project back to I' and m'. 

The development need not again be descrilied in detail Imt is left for the student to construct, 
with the reminder that for the actual distance of any corner of the intersection from an edge of 
either prism he must look to that projection which shows the base of that prism in its true size : 
thus the distance of /' from the edge h' is /*"/". 

428. The intersection of pyramidal surfaces by lines and planes. The iirinciple on ■i\-liirh tlie inter- 
section of pyramidal surfaces by plane -sided or single curved surtaces would be obtained is illustrated 
by Figs. 282 and 283. 

(a) In Fig. 282 the line a b. a'h', is supposed to intersect the given pyra- 
mid. To ascertain s' and t' — its entrance and exit points — we regard the 
elevation a' b' as representing a plane perpendicular to V and cutting the edges 
of the pyramid. Project m'. where one edge is cut, to hi, on the plan of the 
same edge. Obtaining n ami o similarly we have /// n o as the plan of the 
section made by plane a'h'. The plan n Ji meets mno at « and t which tlien 
project back to a' b' at s' and t', the points sought. 

As ab, a'b', might be an edge of a pyramid or prism, or an element of a 
corneal, cylindrical or warped surface, the method illustrated is of general appli- 
cabilitj'. 

(b) In Fig. 283 the auxiliary planes are taken vertical, instead of perpendicular to A' as in the 
last case. 

Tlie plane MX cuts a pyramid. To find where any edge r'o' pierces 
tlie plane MX jjass an auxiliary vertical plane x z through the edge, and 
note .'• and z, where it cuts the limits of MX; project these to .t' and z'; 
draw .i:' z', which is the elevation of the line of intersection of the original 
and auxiliary planes, and note s', where it crosses r'o'. Project -s' back to 
.s on the plan of v' o'. 

If a side elevation has been drawn, in which the plane 
in cpiestion is seen as a line M" X", the height of the points 
of intersection can be obtained therefrom directly. 

429. The intersection of fun qvadrangidar jj/jramids. Let 
one pyramid be vertical; altitude r' z' ; base efgh. having 
its longer edges inclined 30° to V. 

The oblique pyramid. Let s' y', the axis of the oblique 
pyramid, be parallel to \' liut inclined 6° to H, and be at some small distance (approximately v h) 
in front of the axis of the vertical pyramid ; then -^ c will contain tlie \Aa\\ of tlie axis, and also of 
the diagonally opposite edges sa and sc, if we make — as we may — the additional requirement 
that a' e', the diagonal of the base, shall lie in the same vertical ijlane with the axis. 

Instead of taking a sej^arate end view of the olilique pyramid we may rotate its liase on the 
diagonal a' c' so that its foremost corner appears at //" and tlie rear corner at d" . whence b' and d' are 
derived by perpendiculars b" b' and d" d', and then the edges -s'^' and s' d'. For the plans b and 
d use sc as the trace of the usual reference plane, and offsets equal to h' h" and d'd", as i^reviously. 
The angle a' c' d', or <f>, is the inclination of the shorter edges of the base to V. 
The intersection. "Without going into a detailed construction for each point of the outline of 
interpenetration it may be stated that eacli method of the ])receding article is illu.strated in this 





162 



THEORETICAL AND PRACTICAL GRAPHICS. 



^"ig-. as-i 



problem, and that there is no special reason why either should have a preference in any case except 
where by properly choosing between them we may a^-oid the intersection of two lines at a very 
acute angle — a kind of intersection which is always undesirable. 

In the interest of clearness 
only the vhible lines of the inter- 
, section are indicated on tlie plan, 
(a) Auxiliary plane perpendicular 
tn V. To find 'm, the intersection 
of edge .if d with the face v h e, 
take s' d' as tire trace of the 
auxiliary plane containing tlie 
edge in question; this cuts the 
limiting edges of the face at i' 
and n' wliich then project back 
to the plans of the edges at i 
and II. Drawing ni we note m, 
when it crosses s d, and project 
ii) to m' on s' d'. Had ni failed 
to meet s d within the limit of 
the face v h e we would conclude 
that our assumption that .s d met 
tliat face was incorrect, and would 
tlien proceed to test it as to some 
other face, unless it was 
evident on ins})ection 
that the edge cleared 
the other solid entirely, 
^>„ as is the case with s h, 
,,--" / s'h', in the present in- 
stance. By using s' h' 
as an auxiliary plane 
the student will get a 
graphic proof of failure 
to intersect. 

(b) Auxiliary plane 
vertical. This case is 
illustrated by using v g 

as the trace of an auxiliarj' vertical plane containing the edge vg,r'(/'. Thinking this edge may 
possibly meet the face sba we proceed to test it on that assumi^tion. 

The plane vg crosses » a at I, and sb at p; these project to /' on s' a' and to p' on »' b' ; 
then p'l' meets v' g' at </, which is a real instead of an imaginary intersection since it lies between 
the actual limits of the face considered. From q' a vertical to vg gives q. 

The order of obtaining the points. The start may be with any edge, but once under way the 
progress should be uniform, and each point joined with the jDreceding as soon as obtained. Thus, 
supposing that q' was the point first found, a look at the i^lan would show that the edge sa of 




THE IXTERSECTIOX OF SIXdLE CURVED SURFACED. 



163 



the oblique pyramid would be reached before vh on the other, and the next auxiliary plane -n-ould 
therefore be passed through •:< « to find u u' ; then would come vh and -s r/. Running clown from 
m on the face * f/ r we find the positions such that insj^ection ^^■ill not axail. and the only thing to 
do is to try. at random, either a plane through c /; or one through -s- *■ ; and so on for the 
remaining points. 

The dcvrkipmcnU. No figure is furnished for these, a.s nearly all tliat the student requires for 
olitaining them has been set furtli in Art. 396, Case 6. The only additinnal points to which attention 
need lie called are the cases where the intersection falls on a fan' instead of an edge. For 
example, in developing the mi leal pyramid we would find the develoi:)ment of j' by drawing v' j', 
prolonging it to o', and projecting the latter to o, Avhen fxa would be the real distance to lay off 
itom / on the development of the base; then laying off the real length of r' j' on c' o' as seen in 
the development we would have the point sought. Similarly, with tt', draw v x ; make v.,x., = v.r^ 
and r.^ (.-i = altitude r' z' ; then i\x„ is the true length of rx (in space); also, making r.,t,^^vt and 
drawing t.,tj we find r,;, to lay oif in its proper jilace on the development of the same face vfg. 
430. .4/) elbow or T- joint, the interxertion of two equal eijUnders whose axes meet. Taking up curved 
surfaces the suni^lest case of intersection that can occur is the one under consideration, and which 
is illustrated by Fig. 285. i^igr- sss. 

The conditions are those stated in Art. 423 for a plane intersection, 
which is seen in a'b' and is actually an ellipse. 

The vertical piece appears in plan as the circle /h q. To lay off the 
eqnidixtant elements on each cylinder it is only necessary to divide tlie 
half plan of one into equal arcs and project the points of division to the 
elevation in order to get the full elements, and where the latter meet a' 1)' 
to draw the dotted elements on the other. 

The derehipment of the horizontal cylinder is shown in the line-tinted 
figure. The curved boundary, which represents the develoi^ed elliiise, is in 

reality a sinusoid 
(Refer to Art. 171). 
The relation of 
the developed ele- 
ments to their 
originals, fully dc- 
.scribed in Art. 120. 
is so evident as to 
require no further 
remark, except to 
call attention again to the fact that their 
distances apart, f,./",, ,fi.'/,, etc., ecj[ual the 
reetlfieation of the small arcs of the plan. 
431. To turn a right angle with a jiij)e by 

J f/, 1 a four-piece elbo/w. Supposing that the blast 

pipe of a furnace was ti> be carried around a bend by a four-piece elbow; the procedure would 
still closely resemble that of the last problem. Instead of one joint or curve of intersection there 
would Vie three, one less than the number of pieces in the pipe. 

Let oq'< sliow the size of the cylinders employed, and be at the same time the plan of the 





164 



THEOBETICAL AND PRACTICAL GRAPHICS. 



vertical piece o' s' n' a'. Until we know where a' n' will lie we have to draw o' a' and s' n' until 
they meet the elements from <S" and T', and get the joint m M' as for a two-piece elbow. On 
mM' produced take some point v', use it as a centre for an arc t' x y t" tangent to the extreme 
elements; di^dde this arc, between the tangent points, into as many equal parts as there are to be 
pieces in the turn; then tangents at x and j/— the intermediate points of division— will determine 
the outer limits of the joints at a', b' and -7. Draw a' v', finding n' by its intersection with s s' ; 
then n'l' parallel to a'b', and similarly for the next piece. 

The developments of the smaller pieces would be equal, as also of the larger. One only is 
shown, that of a'n'l'b', laid out on the developed right section on v' x. The lettering makes the 
figure self- interpreting. 

432. The intersection of two cylinders when each is partially exterior to the other. The given con- 
dition makes it evident, by Art. 423, that a continuous non- plane curve will result. 

Let one cjdinder be ver- 
^^^-^^'^- tical, 2" in diameter and 2" 

high. This is shown in half 
plan in h k I, and in front and 
side elevations between hori- 
zontals 2 " apart. 

Let the second cylinder be 
horizontal; located midway be- 
tween the upper and lower 
levels of the other cylinder; 
its diameter -f". On the side 
elevation draw a circle a" b" 
c" d" of I" diameter, locating 
its centre midway between k" I" 
// and k ^ N', and in such posi- 
tion that a." shall be exterior 
to k"k,. The elevation of the 
horizontal cylinder is then pro- 
jected from its end view and 
is shown in jjart without con- 
struction lines. 
The curiae of intersection is obtained by selecting particular elements of either cylinder and noting 
where they meet the other surface. 

The foremost element of the vertical cylinder is k...k'n'm'. Its side elevation, k" k^, meets 
the circle at n" and m" which give the levels of n' and m' respectivel3^ 

On the horizontal cjdinder the highest and lowest elements are central on the plan and meet 
the vertical cylinder at e, which jDrojects down to the elements d' and b'. 

The front and rear elements, c and a, would be central on the elevation. The -^-ertical line 
drawn from the intersection of element c with the arc hkl gives the right-hand point of the curve 
of intersection, at the level of a'. 

Any element as g x may be taken at random, and its elevation found in either of the following- 




ways: (a) Transfer gs, the distance of the element from, Jf iV, to 



on the side elevation, and 



draw xg" and g"y', to which last (prolonged) project g at g'; or (b) prolong g x io meet a 



THE INTERSECTION OF SINGLE CURVED SURFACES. 



165 



semi -circle on ac at g'" ; make a' y' = x g'" and draw y' g'. The same ordinate xg'", if laid off 
behiv a, would obviously give the other element which has the same plan gx, and to which g 
projects to give another point of the desired curve. 

433. The inUrscction of a vertical cone by a horhontal prism. Let the cone have an altitude, ww' 
of 4"; diameter of base, 3". (As the cylinder is entirely in front of the axis of the cone only one- 
half of the latter is represented.) ng-. ebb. 

For the cylinder take a diameter of 
length S^"; axis parallel to Y, f" 



4" 

s 

above the base of the cone, and |" from 
the foremost element. Draw )i s parallel to 
p' r' and f" from it; also g' m' horizontal 
and -f" from the base; their intersection 
s is the centre of the circle a" d' c" m', of 
■|" diameter, which bears to the element 
■p' r' the relation assigned for the cylinder 
to the foremost element; said circle and 
p'wio' are thus, practically, a side elevation 
of cylinder and cone, superposed upon the 
ordinary view. 

The dimensions chosen were purposely 
such as to make one element of the cone 
tangent to the cylinder, that the curve 
of intersection might cross itself and give 
a mathematical " double point." 

The width d h, of the jolan of the 
cylinder, equals »; ' d'. The plan of the 
axis (as also of the highest and lowest 
elements, a' and c') will be at a distance 




s g' from u\ Any element as x' y' h' is shown in plan parallel to p q, and at a distance fi-om it equal 
either to h' y' if on the rear or to h' x' if on the fi'ont. 

The element through v, on wliich /' falls, is not drawn sejjarately from bf in jalan, since vf 
and m' g' are so nearly equal to each other; but / must not be considered as on the foremost 
element of the cylindei; although it is apparently so in the plan. 

For the intersection j)ass auxiliary horizontal planes through both surfaces; each will cut from 
the cone a circle whose intersection with cylinder-elements in the same plane will give i3oints sought. 

A horizontal plane through the element a' would be represented by a'o'., and would cut a circle 
of radius o's' from the cone. In plan such circle would cut the element a at point 1, and also at 
a point (not numbered) sj^nmetrical to it with respect to w Q. Similarly, the horizontal plane through 
the element x' h' cuts a circle of radius I' h' from the cone; in jalan such circle would meet the 
elements x and y in two more points (5 and 8) of the curve. 

As the curve is symmetrical with respect to tvQio' the construction lines are given for one -half 
only, lea^dng the other to illustrate shaded effects. The small shaded portion of the elevation of the 
cylinder is not limited by the curve along which it would meet the cone, but by a random curve 
■which just clears it of the right-hand element of the cone. 

434. To find the diameter and inclination of a cylindrical pipe that will make an elbow ivith a conical 



166 



THEORETICAL AND PRACTICAL GRAPHICS. 



pipe on a given plane section of the latter. Let vab be a vertical cone, and c d the ellii^tical i^lane 
section on which the cylindrical piece is to fit. The diameter of the desired cylinder will eqvxal 
the shorter diameter of the elliiDse c d. To find this bisect c d at e ; draw fh horizontally 

through e, and on it as a diameter draw the semi- 
circumference fgh; the ordinate e g is the half width of 
the cone, measured on a perpendicular to the paper at e, 
and is therefore the radius of the desired cylinder. 



E-igr- sss- 






/ 



Ofig-. aso. 





In Fig. 290, the base NG equals twice 
ge of Fig. 289. At first indefinite perpen- 
diculars are erected at N and G, on one of 
which a point C is taken as a centre for 
an arc of radius equal to c d in Fig 289. 
The angle <^ being thus determined is next 
laid off in Fig. 289 at c, and cdN"G" 
made the exact duplicate of G D N G, com- 
pleting the solution. 

The developments are obtained as in 
Arts. 120 and 191. 

435. To determine the conical piece which 
will properly connect two unequal cylinders of circular section, ivhose axes are parallel, meeting them either 
(a) in circles or (b) in ellipses; the pjlanes of the joints being parcdlel. 

(a) TFAot the joints are circles. To determine the conical frustum b ehc jjrolong the elements e b 
and h c to v ; develop the cone v ...eh as in Art. 418, and on each element as seen in the develop- 



Thus the element whose plan is v^k 



X^igr- 2©1_ 



ment laj' off the real distance from v to the upj)er base h c 
is of actual length vlc^ and cuts the upi^er base 
at a distance v n from the vertex, which distance 
is therefore laid on vh^ wherever the latter aj)- 
pears on the development. 

(b) When the joints are ellipses. Let the 
elliptical joints no and qr be the bases of the 
conical piece qno r. To get the development 
complete the cone by prolonging qn and or to 
w; prolong qr and drop a perpendicular to it 
from w; find the minor axis of the ellipse qr -X. 
as in the first jjart of Art. 434 and having con- 
structed the elliijse proceed as in Art. 418, since 
in Fig. 255 the arc abc .. .g is merely a special 
case of an elliiDse. 

436. The projections and patterns of a bcdh-tub. Before taking up more difiicult problems in the 
intersection of curved surfaces one of the most ordinary applications of Grajjhics is introduced, partly 
by way of illustrating the fact that the engineer and architect enjoy no monopoly of practical 
projections. 

In Fig. 292 the height of the main portion of the tub is shown at a'd'. Let it be required 
that the head end of the tub be a portion of a verticcd right cone whose base angle c' b' a' equals the 
flare of the sides, such cone to terminate on a curve whose vertical projection is o'n'z'a'. Draw 




THE lyTERSECTIOX OF SINGLE CURVED SURFACES. 



16t 



^ief. 2SS. 



two lines, b'V and c'/', at first indefinite in length and at a distance a' d' apart. Take a' d' vertical, 

and regard it not only as the projection of the elements of tangency of the flat sides with the conical 

end, but also as the elevation of part of 

the axis, prolonging it to represent the 

latter. Use v, the plan of the axis, as the 

centre for a semicircle of radius c r, whose 

diameter e d is the width of the liottoni of 

the tub. Project c to c'; make angle v'c'd' 

equal to the predetermined flare of the 

sides; prolong v' c' to b' and o' ; jjroject 

b' to b on vc i^rolonged and draw arc 

a b m with rachus b v, obtaining a m for the 

width of the plan of the tojj. 

The plan of one -half the curve o' n' z' a' 
is shown at onzm and is thus found: 
Assume any element v' x'y'; jjrolong it to 
z'; obtain the plan vxy and project s' upon 
it at s. Similarly for n and as many inter- 
mediate points as it might seem desirable 
to obtain. 

Assuming that the foot of the tub is 
composed of an oblique cone whose section, /(/«, with the bottom is equal to ecd, and whose base 
angle is h' i' k\ we project / to /', draw /'/:' at the given angle to the base, jDroject h' to Ic, and 
through the latter draw the semicircle r I q with radius b i\ obtaining the plan of the upper base. 

Joining the tangent points ;• and s, h and q, we have rs and hq as the elements of tangency 
of sides with end. Their elevations coincide in h'l', which meets k' i' at v", whose plan is v^ on hq. 

I^ig-. SS3. 





'The development. Fig. 293 is the development of one-half of the tub. EM equals b'c'; VO 
equals v'o'; YZ equals v'z", the true length of v' z' , obtained, as in previous constructions, by car- 
rj'ing z to s,, thence to level of z' . Similarly at the other end. (Reference Articles 191, 408, 418.) 

437. The intersection of a vertical cylinder and an oblique cone, their axes intersecting. 

Let MBd and M' R' P' X' be the jsrojections of the cylinder; v'.a'b' and v.anbm those of the 
cone. The axes meet o' at an angle 6 which is arbitrarv. 



168 



THEORETICAL AND PRACTICAL GRAPHICS. 



S" g-. 2S-S:. 



The ellipse anbm is supposed to be constructed b}^ one of the various methods employed when 
the axes are known; and in this case we get the length of m n from a' b' and its jMsition from n', 
while a 6 is vertically above a'b'. 

(a) Solution by auxiliary vertical planes. Any vertical plane 
vis will cut elements from the cylinder at e and I; also, 
from the cone, elements which meet the base at s and t. 
Project s and t to s' and t', join the latter with the vertex 
v' and note V and e' (just below d') where ihej cross the 
vertical projection of the elements from I and e; these will 
be points in the desired curve of intersection. 

By assuming a sufficient number of vertical planes through 
V the entire curve can be determined. 

(b) Solution by auxiliary spheres. If two surfaces of revo- 
lution have a common axis they will intersect each other in 
a circle whose plane is perpendicular to that axis.* This 
property can be advantageously applied in problems of inter- 
section. 

With o' — the intersection of the axes — as a centre, we 
may draw circles with random radii o'f, o'i, and let these 
represent spheres. The sjihere f g'w intersects the cone in the 
circle f g'; the cylinder in the circle h'h'. These circles inter- 
sect each other at x in a common chord whose extremities are 
points of the curves sought. 
They are both projected in the 
point X. 

A second pair of circu.lar 
sections, lying on the same 
auxiliary siDhere, are seen at 

pq and rio, their intersection 2 being another point in the solution. 

The iDoint y results from taking the smaller sphere. 

438. Intersection of a cylinder and cone, their axes not lying in 

the same plane. 

In Fig. 295 let the cjdinder be vertical and the cone oblique, 

the axis of the latter being jjarallel to V and inclined 0° to H, and 

also lying at a distance x back of the axis of the cylinder. 

The auxiliary surfaces employed may preferably be vertical planes 

through the vertex of the cone, since each will then cut elements from 

both cjdinder and cone. Thus, vfe is the h. t. of a vei-tical plane 

which cuts e v, e'v' from the cone, and the vertical element through 

/ from the cylinder; these meet in vertical projection at /', one point 

of the desired curve. The plan of the intersection obviously coincides 

with that of the cylinder. 




X'ig-. sss. 




*By the deflnitlon of a surface of revolution (Art. 340) any point on it can generate a circle about its axis. If, then, two 
surfaces have the same axi^t any point common to both surfaces Tvould generate one ancl the same circle, which must also lie 
on both surfaces and therefore be their line of intersection. 



THE INTERSECTION OF SINGLE CURVED SURFACES. 



169 



h! 



'\9 



X'ig'- ESS. 




439. Conical elbow; right cones meeting at a given angle and having an elliptical joint. This is one 

of the cases mentioned in Art. 423 as not admitting of illustration t^ 

in the same way as when dealing with surfaces of uniform cross 

section, but a plane intersection is nevertheless secured as with 

cylinders by making the extreme elements of the cones intersect. 
Let vx in Fig. 296 be the axis of one of the cones, li xyz 

is the required angle between the axes bisect it by the line ym, 

and draw the joint cd jDarallel to such bisector. The right cone 

which is to meet a b c d on c d must be capal^le of being cut in 

a section equal to cd by a plane making an angle with its axis, 

and must ob-\dously have the same base angle as the original cone; 

since, however, the upper portion vdc of the given cone fulfills these conditions we may employ 

it instead of a new cone, rotating 
it about an axis jj t which is per- 
pendicular to the plane of the 
ellipse dc and passes through its 
centre. The point o, in which 
the axis vx meets the plane dc, 
will then ajoj^ear at .s, by making 
o'p=ps; sv', drawn parallel to 
2/2, will be the new direction of 
vo; and an arc from centre d 
with radius cv will give v' , which 
is then joined with d and c to 
complete the construction. 

If the length of the major 
axis of the elliptical joint had 
been assigned, as ef for example, 
that length would have first been 
laid off from some point e on 
the extreme element and parallel 
to ym, then from / a parallel to 
V e, giving g on v c ; then gh^ 
parallel and equal to ef, gives 
the joint in its proper place. 

440. Right cones intersecting in 
a non-plane curve; axes meeting at 
an oblique angle. Let one cone, 
v'.a'b', (Fig. 297) be vertical; the 
other, oblique, its axis meeting 



The jnlane a' b' of the base 
of the vertical cone cuts the other cone in an ellipse whose longer 
axis is e'f. As in Art. 434 determine g' h', the semi-minor axis 
of this ellipse. Project e', g' and /' wp to e, g and /; make 







170 



THEORETICAL AND PRACTICAL GRAPHICS. 



ghj^ and gh^ each equal to g' h' ; then on ef and hj^h.^, as axes construct the ellipse e h^f h.^ as 
in Art. 181. Tangents from v^ to the ellipse complete the plan of the oblique cone. 

(a) The curve of intersection, found by auxiliary ]ilanes. In order that each auxiliary plane shall 
contain an element (or elements) of each cone, it must contain both vertices and therefore the line 
v'v", which joins them; hence its trace on the plane e'a'b' must pass through the trace, t' t, of 
such hne on that plane. Take tx sls the horizontal trace of one of these auxiliary jDlanes. It cuts 
elements starting at i and I on the base of the oblique cone. One of the elements cut from the other 
cone is v 2^, which in vertical projection (v' p') crosses the elevations of the other elements at q' and 
r', two points of the curves sought. Since the extreme elements of the cones are parallel to V we 
will have c' and d' — the intersections of their elevations — for two more points of the curve. Ha^dng 

found other points by repeating the 
same jsrocess the curve c'c/rd' is 
drawn through them, and the cones 
may then be developecl as in Art. 191. 
(b) Method by auxiliary spheres. 
Since the axes intersect we may use 
auxihary sjaheres as in Case , (b) of 
Art. 437. Thus, with o' — the intersec- 
tion of the axes — as a centre, take any 
radius o ' k and regard arc I y s as rejD- 
resenting a j^ortion of a sphere which 
cuts the cones in ks and y z. These 
meet at w, one point of the curve of 
intersection c' q' d'. 

441. Intersecting cones, bases in the 
same plane hut axes not. Let v.kbfg 
and e.sQhj be the j)lans of the cones; 
v.'p'd' and e.'Q'c' their elevations. 

As argued in Case (a) of the last 
j)roblem, the auxiliary planes must con- 
tain the line joining the vertices; their 
H- traces would therefore, in the gen- 
eral case, pass through the trace of 
that line upon the plane of the bases; 
but, in the figure, both vertices hav- 
ing been taken at the same height 
above the bases, the line which joins 
them must be horizontal, hence parallel 
to the H- traces of the auxiliaries: that 
is, XY, ST, QR, etc., are parcdlel 
to V e. 

It happens that the trace MN of 
the foremost auxiliary j)lane is tangent to both bases, hence contains but one element of each cone 
and determines but one point of the desired curve. These elements, a e and b v, meet at n, while 
their elevations intersect at n'. 




THE INTERSECTION OF SINGLE CURVED SURFACES. 171 

Each of the other planes, excejit X Y, being secant to both l:iases, will cut two elements from 
each cone, their mutual intersections giving four points of the curve of interpenetration. Thus, in 
plane P, the element e meets v k in g and v d in i; while element h e gives I and m on the 
same elements. 

The plane A' Y lieing tangent to one base while secant to the other gives but two points on the 
curve sought. 

Oi-der of connecting the points. Starting with any plane, as M N, we may trace around the bases 
either to the right or left. Choosing the former we find, in the next jjlane, the point h to the 
right of n on one base, and cl similarly situated with respect to b on the other ; therefore m, on h e 
and d v, is the next point to connect with n. Elements o e and / v give the next point, then u e and 
g V locate s, after which those from j and iv give the last before a return movement on the base of 
the I- -cone. As nothing new would result fi-om retracing the arc gfd we continue to the left from 
ic, although compelled to retrace on the other base, since planes beyond j would not cut the t^-cone. 
The element u e is therefore taken again, and its intersection noted with an element whose projection 
happens to be so nearly coincident ^'s-ith vx that the latter is used. 

Continuing along arcs o ch and ikb we reach the plane M N again, the curves ilx and q n m 
crossing each other then at n — the point lying in that j^lane. Such j)oint is called a double point, 
and occurs on non- plane curves of intersection at whatever point of two intersecting surfaces they 
are found to have a common tangent plane. 

Tracing to the left from cl and to the right from b the elements e and cl v are reached, in the 
plane OP. Their intersection x is joined with n on one side and with the intersection of Se and 
gv on the other. Soon the tangent plane X Y is again reached and a return movem.ent necessitated, 
during which the arc XSQOa is retraced, while on the other base the counter-clockwise motion 
is continued to the initial point b, completing the curve. 

Visibility. The visible part of the intersection in either view must obviously be the intersection 
of those portions of the surfaces which would be visible were they separate, but sunilarly situated 
with resjiect to H and N. 

In plan the j^oint n lies on visible elements, and either arc passing through it is then ^dsible 
till it passes (becomes tangent to, in projection) an element of extreme contour as at m or t, when 
it runs from the upper to the under side of the surface and is concealed from view. 

The point w would be visible on the t;-cone but for the fact that it is on the under side of 
the e-cone. 

A similar method of inspection will determine the visible portions of the vertical projection of 
the curve, which -ndll not be identical with those of the plan. In fact, a curve wholly visible in one 
view might be entirely concealed in the other. 

442. The intersection of a vertical cylinder and an oblique cone, their axes in the same plane. If in 
Art. 440 the vertex v' were removed to infinity the w-cone would become a vertical cyhnder; the 
line v' v" would become a vertical hne through v" ; t would be vertically above ■;;"; but the method 
of solving would be unchanged. 

443. In general, any method of sohT.ng a problem relating to a cone will ap)p)ly with equal 
facility to a cj'hnder, since one is but a special case of the other. The line, so frequently used, that 
passes through the vertex of a cone in the one problem is, in the other, a parallel to the axis of 
the cylinder. Planes containing both vertices of cones become planes parallel to both axes of cylinders. 

In ^'iew of the interchangeal^iUty of these surfaces it is unnecessary to illustrate by a separate 
figure all the jjossible variations of problems relating to them. 



172 



THEORETICAL AND PRACTICAL GRAPHICS. 



444. Intersection of two cones, two pyramids, or of a cone and a pyramid, luhen neither the bases nm- 

axes lie in one plane. 

One method of solving this problem has been illustrated in Art. 429, where the intersection was 

found by using auxiliary planes that 
were either vertical or perpendicular to 
V; we may as easily, however, employ 
the method of the last problem, viz., 
bjf taking auxiliary planes so as to 
contain both vertices. This will be 
illustrated for the problems announced, 
by taking a cone and pyramid; and, 
for convenience, we will locate the sur- 
faces so that one of them will be ver- 
tical, and the base of the other will be 
perpendicular to V, since the problem 
can always be reduced to this form. 

Let the cone v'.a'b', v.cdB, (Fig. 
299) be vertical, and the pyramid o'.r' 
q' p)', o.rqp, inclined. 

We will assume that the projec- 
tions of the pyramid have been found 
as in jjreeeding problems, from assigned 
data, using oo,_, o' p', (taken perpen- 
dicular to the base r' q') as the refer- 
ence line. 

Join the vertices by the line v' o', 
V 0, and prolong it to get its traces, 
ss' and tt', upon the planes of the 
bases. All auxiliary planes containing 
the line v o, v' o', must intersect the 
planes of the two bases in lines pass- 
ing through such traces. . 

Prolong r' q' to meet the plane 
a' b' at X. Project up from X, get- 
ting yz for the plan of the intersection 
of the two bases. 

We may assume any number of 
auxiliary planes, some at random, but 
others more definitely, as those through 
edges of the pyramid or tangent to the 
cone. Taking first one through an 
edge, as or, we have trz for its trace 

■on the pyramid's base, then zs for its trace on H. The elements cv and dv which lie in this 

plane meet the edge or at e and /, giving two points of the curve. These project to o' r' at 

e' and /'. 




INTERSECTIONS,— PRACTICAL ENGINEERING DESIGNS. 



173 



The plane s y, tangent to the cone along the element u v, has the trace // 1 on the base of the 
pjTamid, and cuts lines jo and I- o from its faces. These meet i- «. at two more points of the curve, 
their elevations being found by projecting j to j' and k to k', drawing o'j' and o'k', and noting 
their intersections with r'n'. To check the accuracy- of this construction for either point, as I, draw 
I't'i perpendicular to ru and equal to v'u', join r, with «, and we have in vv^u the rabatment of 
a half section of the cone, taken through the element vu and the axis; then 11^, parallel to vvi, 
will be the height of /' above the base a'b'. 

With one exception, any auxiliary plane between sy and sz will give four points of the inter- 
section. The exception is the plane .s Y, containing the edge o q, and which, on account of hap- 
pening to be vertical, requires the following special construction if the solution is made wholly on 
the plan: Rabat the plane into H; the elements it contains will then appear at Av, and Bv^, 



while the edge o q will be seen in 



(bv makinir 



-d'O, and qqi^= q'Q); elements and edge 



then meet at ./, and T, which counter -revolve to / and N. We might, however, get elevations 
first, as J', by the intersection of element A' v' with edge '/'/; then J from J'. 

In the interest of clearness several lines are omitted, as of certain auxiliary planes, hidden por- 
tions of the ellipses, and the curves in which srq (the rear face) cuts the cone. The student 
should supply these when drawing to a larger scale. 

BRIDGE POST CONXECTIOXS. — GEARING. — SPRINGS. — BOLTS, SCREWS AND NUTS. 



44-5. Detail of a Bridge. — Upper -Chord Post - Connection. — A bridge or roof truss is an assemblage 
of jjieces of iron or wood, so connected that the entire combination acts like a single beam. Figs. 
300 and 301 are what are called "skeleton diagrams" of bridge trusses, each piece or "member" of 
the truss being represented by a single line. A BCD and A' B' C D' are the trusses proper, the 




■S^S- 301. 



A/ I \| M/ \|/ \1/ \|/ [/ I \D 



former being for an overhead track and the latter for a roadway running through the bridge. In 
each case the upper part — called the njqjer chord— (AD, B'C) sustains compre-mon, and is made of 
-built beams," formed by riveting together various plates and lengths of structural iron in such 
manner as to form one practically continuous column. 

The hirer chords (B C, A' D') su.stain tension, and are made of bars of high tensile strength. 

The members that connect the chords are called either ties or struts according as the strain in 
them is tensile or compressive. Collectively they form the ireh of the truss. 

In the form of truss illustrated — which is only one of manj- which have commended themselves 
to the profession — the vertical pieces or "posts," Be, fr/, etc., sustain compression, and are therefore 
"built" columns. They divide the trapezoid into parts called pnnels, which has given the name 
panel gystem to this largely -employed arrangement of bridge members. 

All the diagonal members in both figures, excepting A'B' and CD', are tension bars or rods. 
Bh and C-s are struts whose sole office is to keep the posts Ah and D s vertical; said posts then 
conveying to the masonry whatever weights are transmitted through the truss to A and D respectively. 



174 



THEORETICAL AND PRACTICAL GRAPHICS. 




Fig. 302 is a perspective view of the connection at A between the post A h, the upper chord 
A D, and the four diagonal bars that are projected in A B. The working drawings required for such 
connection are shown in the three views on the opposite page. Three analogous views would be 
required for the connection at the foot of the same post. 

When — as in the case from which our example is taken — there are two railroad tracks overhead, 
^-xir- 303. ^jjg members of the middle truss will usually have different propor- 

tions from those in the outside trusses, and a separate set of three 
views has therefore to be made for each of its post connections, 
so that the smallest number of shop drawings for one such bridge — 
after making all allowance for the symmetry of the structure with 
reference to the central plane MN — would consist of twenty such 
groups of three as are illustrated by Fig. 304. 

The upper projections (Fig. 304) are obviously a front and a 

side elevation. The lower figure may preferably be regarded as a 

plan of the object inverted, since that conception is somewhat more 

natural than that of the post in its normal position, while the 

draughtsman lies on his back and gazes up at it from beneath. 

Fig. 303 shows the inverted plan on a somewhat smaller scale, and, although presented mainly 

to illustrate the contrast between views with shade lines and without, contains one or two serviceable 

dimensions that are omitted on the other plate. 

446. General description. Referring to the wood cut as well as the orthographic projections, we 
find the upper chord to be composed of a long cover-plate, 18" X t^", riveted to the top angles of 
two vertical channel bars set back to back; each channel being 15" high and weighing 200 pounds 
per yard. The cross section of the upper chord is shown in solid black, with just enough space 
intended between plate and channels to show that they are not all in one piece. 

Perpendicular to the vertical faces of the channels and through holes cut therein runs a cylinder 
called a "pin," 4" in diameter and 211" x'lgr. soa. 

" between shoulders" (as marked on the 
plan), that is, between the planes where 
the diameter is reduced and a thread 
turned, on which connection can be made 
with the corresponding post in the next 
truss. 

Four diagonal bars are sustained by 
the pin, the latter passing through holes, 
called "eyes," in the bars. Two of the 
bar heads are between the channels. 

Two plates are inserted between each 
of the outside bars and the nearest channel, 
not only to prevent the bar from touching 
the angle, as at h, but also to relieve the metal nearest the pin from some of the strain. 




The 



longer plate mnFE is next the channel. The other, nmop, has a kind of hub cast on it which 
rounds up to the bar head, as shown in the side elevation. 

The vertical post is made up of an I-beam and two channels, as shown by the black sections 
on the plan. 



UPPER-CHORD POST-CONNECTION. 



175 



-14" 



is'x Vk cover plate 



-vi-H- 




X 



S" 



Railroad Bridge 
Past CnnnBGtinn 

Upper Clinrd, 



When drawing the above the student should make horizontal dividing lines in all fractions. A brush tint of Prussian blue for all the metal 
parts will enhance the appearance of the di-awing very muteriallj-, but the previous lining should be, obviously, in best waterproof ink. 
Centre, dimension, and extension lines should be in continuous red lines, unless for blue printing, in which case all lines will be black. 



176 



THEORETICAL AND PRACTICAL GRAPHICS. 



Between the upper chord and the top of the post is a three -quarter -inch plate, seen best on 
the plans at iflt. It is nicked out 4", near the nuts K, so as to clear the two middle bars S which 
come between the channels. 

A 5 " X 3 " angle - iron runs from outside to outside of channels, and is held by rivets and by 
the bolts marked H. A shorter piece of the same kind is fastened by bolts K to the plate, and 
by rivets to the web of the I-beam. 

447. Hints as to drawing the bridge post connection. Draw the main centre lines first; then the 
plan and side elevations simultaneously, as the horizontal centre line of the plan represents the 
same vertical reference -plane as the vertical centre line of the side elevation, and one spacing of 
the dividers may be made to do double work. 

The solid sections should be drawn first of all; then the pins, bars, and cap plates of the post 
in the order named. The parts already drawn should next be represented on the front elevation by 





3iK3l 

! TEE BAR. 

-I" 

tP 28^ LBS, 

PER YARD. 




15 CHANNEL. 

200 LBS. 
PER YARD. 



projecting up from the plan and across from the side view. The filler plates, mp and m F, with 
their rivets, come next on the front elevation, from which they project to the side. 

Next in order draw the angle irons on the front elevation, with their bolts, H' and K', and 
project them both across and down. Finally put in all remaining rivets, and dimension the views. 

The angle whose bolts are marked H terminates exactly on the edges of the channels, as shown 
in the wood -cut, rather than as indicated in the side elevation. 

448. Structural Iron. In Figs. 305 and 306 the forms of iron more generally used in bridge and 
house construction are shown in cross section, and may advantageously be drawn on an enlarged scale. 

Treated as described in Art. 75 they may be worked up with brush or pen like Fig. 186. 



TOOTHED GEARING. -HELICAL SPRINGS. 



177 




449. Toothed Gearing. When two shafts are to be rotated and a 
constant velocity ratio maintained between them, it is customary to fix 
upon them toothed wheels whose teeth are so proportioned that by 
their sliding action upon each other they produce the motion desired. 
It is not within the intended scope of this work to go at length into 
the theory of gearing, for which the student is referred to such special- 
ized treatises as those of Grant, Robinson, MacCord, Weisbach and Willis; 
but the draughtsman will find it to his advantage to be familiar with 
the following rapid method of drawing the outlines of the teeth of 
a spur wheel, in which a remarkalily close approximation is made by 
circular arcs to the theoretical involute outlines now so much employed. 

450. CN (in Fig. 307) is the radius of the pitch circle, that is, the 
circle which passes through the middle of the working part of the tooth. 

The working outlines outside the pitch circle are called faces (fg, h t), 
while ivithin they are designated as flanks. The flanks are rounded off 
into the root circle by small arcs called fillets. 

The limits of the teeth on the addendum circle, as a, g, h, m, are 
called their points. 

On the pitch circle the distance h i, between corresponding points 
of consecutive teeth, is called the circular pitch (usually denoted by P). 

Knowing the pitch and the number {N) of teeth, the radius of the 
pitch circle will equal PxN-t-^tt. 

As one inch pitch and twenty teeth are taken as data for the illus- 



tration, we have CiV= 3".18+. 

The other jDroportions are also expressed 
in terms of the pitch, a frequently -used 
system therefor being indicated on the 
figure. 

If i is a point through which a tooth 
outline is to pass, draw C'i, and on it as 
a diameter describe the semi -circumference 
Csi. An arc from centre i, with a radius 
of one-fourth Ci, will give the centre s 
of the outline hij. 

Draw the "circle of centres" through 
s, from centre C. Then with s i in the 
dividers, and from centre / find q, which 
use in turn for arc g f e, and so continue. 
The loidth of rim, v to, is often made, 
by a "shop" rule, equal to three-fourths 
the pitch. Reuleaux gives for it the fol- 
lowing formula : u if = 0.4 P -)- .1 2. 

Diametral pitch is very frequently used 

instead of circular pitch, and is simply 

451. Helical Spirings. Draw first (Fig. 



E'igr- SOT. 




Pitch (P) = bi = fl 
Depth, Mv, = ,7P 
Working Depth, Mz,=|.6P 



Addendum, M N, = .3 (? 



Width of Tooth, fi,=-n 
Width of Space, i\r^, 
BacUlashrn 
Clearance, zv,= io 



the number of teeth per inch of pitch -circle diameter. 
308) a central helix acfm..T, as follows: Divide a a, — 



178 



THEORETICAL AND PRACTICAL GRAPHICS. 



ngr- 309. 



T^ig-. 30S- 





which is the pitchy or rise in one turn — into any number of equal parts, and the semi -circumference 
AEM into half as many equal divisions; then each point marked with a capital on the half plan 
gives two elevations (denoted by the same letter small) by a process which is self-evident. 

If the spring is circular in cross- section draw 
a series of circles having centres on the helix, and 
whose diameters equal that of the spring; then 
the outlines of the spring will be curves that are 
tangent to the circles. 

If the spring he 
small the curvature of 
the helix may be ig- 
nored, and a series of 
parallel straight lines 
employed instead, drawn 
tangent to circular arcs as in Fig. 309. 

The upper half of the figure gives 
the method of construction, while the 
lower shows the spring in section, and 
surrounding a solid cylindrical core. 

452. Springs of rectangular cross -section. 
Fig. 311 shows a spring of this descrip- 
tion, formed by moving the rectangle abed helically, each point describ- 
ing a helix which can be constructed as described in the last article. 

When any considerable number of turns of the same helix has to 
be drawn it will save time if the draughtsman will shape a strip of 
pear -wood into a templet, i.e., a piece whose outline conforms to a line 
to be drawn or an edge to be cut, using it then as a curved ruler to 
guide his pen. This is the preferable method for all large work. 

x-ig. sii. 453_ Square -threaded screws. If in- 

stead of spirally twisting a rectangu- 
lar bar the same kind of surface be 
cut upon a cylinder of wood or metal, 
we shall have a square -threaded screw. 
This is illustrated by the upper part of 
Fig. 310, and its construction is self- 
evident after what has preceded. On a larger scale the curva- 
ture of the helices would have to be indicated. 

The upper view is an elevation of a small double -threaded 
square screw, generated by winding two equal rectangles simul- 
taneously around the axis. 

The central figure is an elevation of a single -threaded screw. 
The lower figure is a sectional view of the nut for the single- 
threaded screw, and evidently presents a surface identical with 
that of the back half of the screw which fits it. 





SCREW-THREADS.— BOLTS. — NUTS. 



179 





"f 30; 



-i„ 



b6lt 



-d-r- 



454. Triangular -threaded screws. — United States Standard. — The proportions devised by Mr. William 
Sellers of Philadelphia have been so generally 
adopted as to be known as the United States 
Standard. They are given in the table on the 
3iA next page. 

i Fig. 312 shows a section of the Sellers 

I screw. It is blunt on the thread, and also at 
i the root. The part j) B which is removed 
jg from the point may be regarded as filled in 
at Nst. AB being the pitch (P), the widths 
op, st, are each one -eighth of P. 

With N equal to the number of threads 
per inch, and D the outside diameter of the 
screw or bolt, the value of d — the diameter at 
the root — may be obtained from the formula 
d = D- (1.299 ^N). 
Other proportions are as follows: The pitch is equal to 
0.24 v/ 25+0625 — 0.175. The depth of thread equals 0.65 P. For bolts 
and nuts, whether hexagonal or square, the "width across flats," or 
shortest distance between parallel faces, equals 1.5 Z), plus one- eighth 
of an inch for rough or unfinished surfaces, or plus one -sixteenth of 
an inch for "finished," i. e., machined or filed to smoothness. 

The depth of nut equals the diameter of the bolt, for "rough" 
work. Tables should be consulted for the proportions of finished pieces. 
Fig. 313 is a drawing, to reduced scale, of a finished |" bolt. 
The elevations show a bevel or chamfer, such as is usually given 
to a finished bolt or nut. On the plans this is indicated by the 
circles of diameter p q, the latter usually a little more than three- 
fourths of the diameter a d. 

To draw the lines resulting from chamfering proceed thus: On 
a \'iew showing "width across flats," as that of the nut, draw the 
chamfer lines z u, i, at 30° ^'■s- s^^- 




FLg. 315. 



rig-. 31S. 




to the top, and cutting off 
the desired amount. Draw 
circles on the plans, with 
diameter equal to u 0. Pro- 
ject p and q to P and Q, 
and draw Px and Qy at 80° to the 
top. Make Nk on the nut equal to 
ny on the head. On the latter draw a parallel to P Q, 
and as far from it as ou is from v i. The arcs limit- 
ing the plane faces have their centres found by "trial 
and error," three points of each curve being known. 

When drawn to a small scale screws may be 
represented by either of the conventional methods illustrated by Figs. 314, 315 and 316. 





180 



THEORETICAL AND PRACTICAL GRAPHICS. 





DIMENSIONS OF BOLTS 


AND NUTS, UNITED STATES STANDARD ( SELLERS SYSTEM) 






Proportions of Bolt 


Dimensions of Nats 
Rough and Finished 


Dimensions of Bolt Heads 
Rouph and Finished 


Outside 
Diam. 

D 


At Root of Thread 


N= 
Number 

of 
Threads 
per inch. 


Width ( f ) 
of Flat. 


Across 
■ w > 
Flats 


Across 

<"> 

Corners 


Across 
Corners 


Depth 

fUlf 


Across 
• w 
Flats 


Across 

(w) 

Corners 


Across 

<^ 

Corners 


Depth 


lf-1^ 


Diam. 


Area 


of Nut 


of Hea'd 


R 


F 


R 


R 


R 


F 


R 


F 


R 


R 


R 


F 


1 
i 


.185 


.026 


20 


.0062 


1 
2 


7 
16 


37 
64 


7 
10 


1 
4 


3 

16 


1 


7 
16 


37 
61 


7 
10 


1 

4 


3 
16 


6 
10 


.240 


.045 


18 


.0074 


10 
32 


17 
32 


11 
16 


■ 5 
6 


•0 
16 


1 
4 


19 
32 


17 
32 


11 
16 


5 
6 


19 
B4 


1 
4 


3 

8 


.294 


.067 


16 


.0078 


11 
16 


5 

8 


51 
64 


63 
64 


3 

8 


5 
16 


11 
16 


5 
8 


51 
64 


63 
64 


11 
32 


5 
16 


7 
16 


.344 


.092 


14 


.0089 


25 
32 


23 
32 


9 
10 


1I4 


7 
16 


3 

8 


25 
32 


23 
33 


9 
10 


ill 


25 

64 


3 

8 


1 
2 


.400 


.125 


13 


.0096 


7 
8 


13 
16 


1 


li 


1 
2 


7 
16 


7 
8 


13 
16 


1 


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ORTHOGRAPHIC PROJECTION UPON A SINGLE PLANE. 



241 



CHAPTER XV. 

AXONOMETKIC (INCLUDING ISOMETEIC) PROJECTION.— ONE -PLANE DESCEIPTIVE GEOMETKY. 



621. ■\\'hen liut one plane of i^rojection is employed there are but two applications of ortho- 
graphic projection having special . names. These are Axonometric (known also as Axometric) Projection, 
and One-Plane Descriptive Geometry or Horizontal Projection. 



AXOXOMETRIC PROJECTION. — ISOMETRIC PROJECTION. 



622. Axonometric Projection, including its much -employed special form of Isometric Projection, is 



applicable to the rej^resentation of the parts or 
"details" of machinery, bridges or other con- 
structions in which the main lines are in direc- 
tions that are mutually perpendicular to each 
other. 

An axonometric drawing has a pictorial 
effect that is obtained with much less work 
than is involved in the construction of a true 
perspective, yet which answers almost as well 
for the conveying of a clear idea of what the 
object is; while it ma}' also be made to serve 
the additional purpose of a working drawing, 
when occasion requires. 

623. Fundamental Problem. — To obtain the 
orthographic jrrojection of three mutually perpendicu- 
lar lines or axes, and the scale of real to projected 
lengths. Let ab, he and hd (Fig. 394) be the 
projections of three lines forming a solid right 
angle at h. Let the line a b be inclined at 
some given angle 6 to the j^lane of projection. 
Locate a vertical plane parallel to a h and pro- 
ject the latter upon it at a'b', at ^° to the 
horizontal. Since the ])lane of the other two 
axes is perpendicular to ah, a'b', its traces 
will be P'd'R. (Art. 303). 

In order to find either c or d we need to 
know the inclination of the axis having such 
point for its extremity. Supposing j3 given 
for ch, draw b' C at y8° to GL; project C to 



IFig-. 3S-5:. 




centre b, obtaining c. 



Cj and draw arc c^c. 

Join a with c; then a c is the trace of the plane of the axes h a and b c, and being perpen- 
dicular tu the third axis we may di-aw the latter as the line ebd, making 90° with a c. 



242 THEORETICAL AND PRACTICAL GRAPHICS. 

Carry d io d^ about h; project d^ to D and join the latter with h'. Then Dh' is the true 
length, and b' D L (or 40 the inclination, of the third axis, h d. 

Lay off a'n', D s' and Ct', each one inch. Their projected lengths on the horizontal are respec- 
tively a'n, Ds and Ct. The latter are then the lengths, representative of inches, for all lines 
parallel to ab, be and h d resJ)ectivel3^ 

624. To make an axonometric projection of a one-inch cube, to the scale just obtained. 
Although not absolutely necessary it is cwstomary to take one axis verticcd. 

Taking, the a6-axis vertical, ' the cube in Fig. 394 fulfills the conditions. For BA equals a'n; 
B D" equals D s, and EC" equals Ct, while the angles at B equal those at b. 

The light being taken in the usual direction, i.e., parallel to the body -diagonal of the cube 
(C" R), the shade lines indicated are those which separate illumined from unillumined surfaces, and 
are those which could, therefore, cast shadows. 

625. The axonometric projection of a verticcd pyramid, of three -fourths -inch altitude and inch -square 
base, to the same scale as the cube. The pj^ramid in Fig. 394 meets the requirements, xw y z having 
been made equal to C" B D" X; while the altitude m.M, rising from the intersection of the diago- 
nals of the base, equals three -fourths a'n, the inch -representative for the vertical axis. 

626. To draw curves in axonometric projection obtain first the jDrojections of their inscribed or cir- 
cumscribed polygons, or of a sufficient number qf secant lines; then sketch the curve through the 
points on these new lines which correspond to the points common to the curves and lines in the 
original figure. This will be illustrated fully in treating isometric projection. 

627. Isometric Projection. — Isometric Draicing. When three mutually perpendicular axes are 
equally inclined to the plane of projection they will obviously make equal angles (120°) with each 
other in projection. This relation led to the name "isometric," implying equal measure, and also 
obviates the necessity for making a separate scale for each axis. 

The advantages of this method seem to have been first brought out by Prof. Farish of England, 
who presented a paper upon it in 1820 before the Cambridge Philosophical Society of England. 

628. In practice the isometric scale is never used, but, as all lines parallel to the axes are equally 
foreshortened, it is customary to lay off their given lengths directly upon the axes or their parallels, 
the result showing relative position and proj)ortion of parts just as correctly as a true projection, 
but being then called an isometric drawing, to distinguish it from the other. It would, obviously, 
be the projection of a considerably larger object than that from which the dimensions were taken. 

Lines ixirallel to the axes are called isometric lines. 

Any plane parallel to, or containing two isometric axes, is called an isometric plane. 

E'lg-- 3©=.. g29. To make an isometric drawing of a cube of three -quarter -inch edges. 

^^ ^\^ Starting with the usual isometric centre, 0, (Pig. 395) draw one axis vertical, 

and on it lay off OA equal to three -fourths of an inch. OC and OB are 
then drawn with the 30 "-triangle as shown, made equal in length to A, 
and the figure completed by parallels to the lines already drawn. 

One body -diagonal of the cube is perpendicular to the paper at 0. 
630. To draw circles and other curves isometrically, employ auxiliary tan- 
gents and secants, obtain their isometric representations, and sketch the curves 
through the proper points. 
In Fig. 396 we have an isometric cube, and at M 0' P' N the square, which — by rotation on MN 
and by an elongation of MP' — becomes transformed into M P N. The circle of centre S' then 



ISOMETRIC DRAWING. 



243 



becomes the ellipse of centre 8, whose points are obtained by means of the four tangencies d', F, 
E and G, and by making gn equal to gn', hrii equal to h'm', etc. 

631. The isometric circle may be divided into parts corresponding to certain arcs on the original, 
either (1) hy drawing radii from iS" to MX, as those s-ig-. sss. 

through b', c', d', (which maj' be equidistant or not, 
at pleasure") and getting their isometric representatives, 
which will intercept arcs, as bd', d'e, which are the 
isometric views of b'd', d'e'; or (2) bj- drawing a (^ 
semicircle x i y on the major axis as a diameter, letting 
fall perpendiculars to .r y from various jjoints, and 
noting the arcs as 1-2, 2-3, that are included between 
them and which corresjiond to the arcs ij, j k, origi- 
nallj' assumed. 

632. Shade lines on Uometric drawings. AMiile not 
universally adhered to, the conventional direction for 
the rays, in isometric shadow construction, is that of 
the body -diagonal CR of the cube (Fig. 895). This 
makes in projectioii an angle of 30° with the horizon- 
tal. Its projection on an isometrically- horizontal plane 
— as that of the top — is a horizontal line CB; while 
its projection C A, on the isometric representation of a 
vertical plane, is inclined 60 ° to the horizontal. 

633. -To illustrate the principles just stated Fig. 397 
is given, in which all the lines are isometric, with the 
exception of Dz and its parallels, and ST. The drawing of non- isometric lines will be treated in 

the next article, but assuming the 
objects as given whose shadows we 
are about to construct, we may start 
with any line, as Dz. 

The ray Dd is at 30°. Its pro- 
jection d^d is a horizontal through the 
plan of D. The raj' and its projection 
meet at d. As the shadow begins 
where the line meets the plane, we 
have z d for the shadow of D z. This 
gives the direction for the shadow of 
any line parallel to D z, hence for yv, 
which, however, soon runs into the 
shadow of B C. As b is the intersec- 
tion of the ray B b with its projection 
b-^b, it is the shadow of B, and bib 
that of b^B. Then b v is parallel to 

B C, the line casting the shadi)W being j'^rallel to the plane receiving it. 

In accordance with the principle last stated, de is equal and parallel to D E, and ef to EF. 

At / the shadow turns to g, as the ray f F, run back, cuts MG at /', and f'G casts the /(/-shadow. 





244 



THEORETICAL AND PRACTICAL GRAPHICS. 



^igr- 3SS- 




Then gh equals G H, and hhi is the shadow of Hh. The projection jm catches the ray Mm at 

m. Then mf, equal to Mf, completes the construction. 

The timber, projecting from the vertical plane P Q R, illustrates the 60° -angle earlier mentioned. 

Kk' being perpendicular to the vertical plane, its shadow Kl is at 60° to the horizontal, and Klk 

is the plane of rays containing said edge. Its horizon- 
tal trace catches the ray from k' at k. Then nk, the 
shadow of n'k', is horizontal, being the trace of a ver- 
tical plane of rays on an isometrically- horizontal plane. 
The construction of the remainder is self-evident. 

Letting S T represent a small rod, oblique to isometric 
planes, assume any point on it, as u; find its plan, 
!(,; take the ray through u and find its trace w. Then 
Siv is the direction of the shadow on the vertical plane, 
and at r it runs off the vertical and joins with T 

634. Timber framings, drawn isometrically, are illustrated 
by Figures 398 and 399. In Fig. 398 the pieces marked 

A and B show one form of mortise and tenon joint, and are drawn with the lines in the custom- 
ary directions of isometric axes. ^ig-- sss. 

The same pieces are represented 

again at C and D, all the lines 

having been turned through an 

angle of 30°, so that while 

maintaining the same relative 

direction to each other and being 

still truly isometric, they lie dif- 
ferently in relation to the edges 

of the paper — a matter of little 

importance when dealing with 

comparatively small figures, but 

affecting the appearance of a- 

large drawing very materially. 
635. Non-isometric lines. — Angles 

in isometric planes. In Fig. 399 

a portion of a cathedral roof 

truss is drawn isometrically. 

Three pieces are shown that 

are not parallel to isometric lines. 

To represent them correctly we 

need to know the real angles 

made by them with horizontal or 

vertical pieces, and use isometric 

coordinates or "offsets" in laying 

them out on the drawing. 

In the lower figure we see at 6 the actual angle of the inclined piece Mf to the horizontal. 

Offsets, fl and 10, to any point of the inclined piece, are laid off in isometric directions at f'l' 




ISOMETRIC D RAW IN G.~XOX-I SO METRIC LINES. 



245 



■Fi-s' -aoo. 



and I'C, when C'f'l' (or 6') is the isometric view of 6. A similar construction, not shown, gave 
the directions of pieces D and D'. 

j\Iiich depends on the choice of the isometric centre. Had N been selected instead of B, the top 
surfaces of the inclined pieces would have been nearlj' or quite projected in straight lines, render- 
ing the drawing far less intelligible. 

The student will notice that the shade lines on Fig. 399 are located for effect, and in violation 
of the usual rule, it having been found that the best appearance results from assuming the light in 
siich direction as to make the most shade lines fall centrallj' on the timbers. 

636. Non - isometric lines. — Angles not in isometric planes. To draw lines not Ij'ing in isometric 
planes requires the use of three isometric offsets. As one of the most frequent applications of 
isometric drawing is in problems in stone cutting, we may take one such to advantage in illustrating 
constructions of this kind. 

Fig. 400 shows an arched passage-way, in jjlan and elevation. The surface no, r'l'n'o' is verti- 
tical as far as n'o', and conical (with vertex /, C) from 
there to n " o ". The vertical surface on n n is tangent at 
71 ' to the cylinder n'f'e'o". Similarly, mm is vertical to «i', 
and there changes into the cylinder m'g'h'. 

The radial bed b' g' is indicated on the plan (though not 
in full size) by parallel lines at bcfigzb. The bed a'h' is 
of the same form as b'g', being symmetrical with it. 

In Fig. 401 we have an enlarged drawing of the key- 
stone with the plan inverted, so that all the faces of the 
stone may be correctly represented as seen. The isometric 
drawing is made to corresjDond, that is, it represents the 
stone after a 180 "-rotation about an axis perj^endicular to 
the paper. 

The isometric block in which the keystone can be 

h' 



inscribed is shown in 
dotted lines, its di- 
mensions, derived 
from the projections, 
being length, AA^^aa; breadth, AB=^a'b'; height, AO^a'p. 

The top surface a'b' becoming the lower in the isometric, 
reverses the direction of the lines. Thus, a' is seen at A, 
and b' at B. To get D make AU=a'u, then UD=ud'. 
Make C symmetrical with D and join with B, and also D 
with A. WQ equals w'q', for the ordinate of the middle 
point of the arc. 

D E is not an isometric plane, hence to reach E from A 
we make AT=a't; Te"^te', and e"E^ay (the dis- 
tance of e from the plane a 6). 

The remainder of the construction is but a duplication of 
one or other of the above processes. 

The principle that lines that are parallel on the object will also be parallel on the drawing may 
be frequently availed of in the interest of rapid construction or for a check as to accuracy. 











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246 



THEORETICAL AND PRACTICAL GRAPHICS. 




HORIZONTAL PROJECTION OR ONE -PLANE DESCRIPTIVE GEOMETRY. 

637. One- Plane Descriptive Geometry or Horizontal Projection is a method of using orthographic 
projections with but one plane, the fundamental principle being that the space -position of a point 
is known if we have its projection on a plane and also know its distance from the plane. 

Thus, in Fig. 402, a with the subscript 7 shows that there is a point A, vertically above a and 
iFig-. -3:02. ^i seven units distance from it. The significance of b^ is then evident, and to 

show the line in its true length and inclination we have merely to erect peri^en- 
diculars a A and B h, of seven and three units respectively, join their extremities, 
and see the line A B in true length and inclination. 

In this system the horizontal plane alone is used; One- plane Descriptive is 
«7 *= therefore applied only to constructions in which the lines are mainly or entirely 

horizontal, as in the mapping of small topographical or hydrographical surveys, in which the curva- 
ture of the earth is neglected; also in drawing fortifications, canals, etc. 

The plane of projection, usually called the datum or reference plane, is taken, ordinarily, below 
all the points that are to be projected, although when mapping the bed of a stream or other body 
of water it is generally taken at the water line, in which case the numbers, called indices or refer- 
ences, show depths. 

638. A horizontal line evidently needs but one index. This is illustrated in mapping contour 
lines, which represent sections of the earth's surface by a series of equidistant horizontal planes. 




In Fig. 403 the curves indicate such a series' of sections made by planes one yard, metre or 
other unit apart, the larger curve being assumed to lie in the reference or datum plane, and there- 
fore having the index zero. 

The profile of a section made by any vertical plane JfiV would be found by laying off — to any 
assumed scale for vertical distances — ordinates from the points where the plane cuts ^^s- ■ao'S:- 
the contours, giving each ordinate the same number of units as are in the index of 
the curve from M'hich it starts. Such a section is shown in the shaded portion on 
the left, on a ground line PQ, which represents MN transferred. 

639. The steepness of a plane or surface is called its slope or declivity. 
A line of slope is the steepest that can be drawn on the surface. A scale of 
slope is obtained by graduating the plan of a line of slope so that each unit 
on the scale is the projection of the unit's length on the original line. "/ 
Thus, in Fig. 404, if m «. and o B are horizontal lines in a plane, one hav- 
ing the index 4 and the other 9, the point B is evidently five units 
above A, and the five equal divisions between it and A are the projections of those units. 




HORIZONTAL PROJECTION OR ONE-PLANE DESCRIPTIVE. 



247 



E-ig-. 4:OB. 



The scale of slope is often used as a ground line upon which to get an edge view of the 
plane. Thus, if B B' is at 90° to B A, and its length five units, then B' A is the plane, and </> 
is its inclination. 

The scale of slope is always made with a double line, the heavier of the two being on the 
left', ascending the plane. 

As no exhaustive treatment of this topic is proposed here, or, in fact, necessary, in view of 
the simplicity of most of the practical applications and the self-evident character of the solutions, 
only two or three typical problems are presented. 

640. To find the intersection of a line and plane. Let u-i!,b^„ be the line, and XY the plane. 
Draw horizontal lines in the plane at the levels of the indexed 
points. These, through 15 and 30 on X Y, meet horizontal lines 
through a and 6 at e and d; e d is then the line of intersection 
of XY and a plane containing ab; hence c is the intersection of 
the latter with X Y. 

The same point c would have resulted if the lines o e and b d 
had been drawn in any other direction while still remaining parallel. 
Fig. -ios. 641. To obtain the line of inter- 

section of tioo planes, draw two hori- 
zontals in each, at the same level, 
and join their points of intersection. 
In Fig. 406 we have mn and 
qn as horizontals at level 15, one 
in each plane. Similarly, xy and ys are horizontals at level 30. The planes 
intersect in y n. 

Were the scales of slope parallel, the planes would intersect in a horizontal line, one point of 
which could be found by introducing a third plane, oblique to the given planes, and getting its 
intersection with each, then noting where these lines of intersection met. 

642. To find the section of a hill by a plane of given slope. Draw, as in the problem of Art. 640, 
horizontal lines in the plane, and find their intersections with contours at the same level. Thus, 
in Fig. 403, the plane XY cuts the hill in the shaded section nearest it, whose outlines pass 
through the points of intersection of horizontals 10, 20, 30 of the plane, with the like -numbered 
contour lines. 





248 



THEORETICAL AND PRACTICAL GRAPHICS. 



CMAPTEB XVI. 



OBLIQUE OE CLINOGRAPHIC PROJECTION. — CAVALIER PERSPECTIVE. — CABINET PROJECTION. 

MILITARY PERSPECTIVE. 



E'xg-. •5=07. 




643. If a figure be projected upon a plane by a system of parallel lines that are oblique to 
the plane, the resulting figure is called an oblique or dinographic 'projection, the latter term being more 
frequently employed in the applications of this method to crystallography. Shadows of objects in 
the sunlight are, practically, oblique projections. 

In Fig. 407, ABnm is a rectangle and mxyn its oblique projection, the parallel projectors Ax 
and By being inclined to the plane of projection. 

644. When the prelectors make 45° ivith the plane this system is known either as Cavalier Perspective, 

Cabinet Projection or Military Perspective, the plane 
of projection being •\-ertical in the case of the 
first two, and horizontal in the last. 

645. Cavalier Perspective. — Cabinet Projection. — 
Military Perspective. As just stated, the projectors 
being inclined at 45° for the system known b}^ 
the three names above, we note that in this 
case a line perpendicular to the plane of pro- 
jection, as Am or Bn (Fig. 407), will have a 
projection equal to itself It is, therefore, 
unnecessary to draw the rays for lines so situated, as the known original lengths can be directly 
laid out on lines drawn in the assumed direction of projections. 

Let ab c d .n be a cube with one face coinciding with the A'ertical j)lane. If the arrow m indi- 
cates one direction of rays making 45° with V, then the ray hn, parallel to m, will give h as 
the projection of n, and from what has preceded we should have c h equal to c n, and analogously 
for the remaining edges, giving abcd.i for the cavalier perspective of the cube. 

Similarly, EKH is a correct projection of the same cube for another direction of projectors, and 
we may evidently draw the oblique edges in any other direction and still have a cavalier perspec- 
tive, by making the projected line equal to the original, when the latter is perpendicular to the 
plane of projection. 

646. Oblique projection of circles. Were a circle inscribed in the back face of the cube DKG 
(Fig. 407) the projectors through the points of the circle would give an oblique cylinder of rays, 
whose intersection with the vertical plane DX would be a circle, since parallel planes cut a cylinder 
in similar sections. We see, therefore, that the oblique projection of a circle is itself circular when 
the plane of projection is parallel to that of the circle. In any other case the oblique projection of 
a circle may be found like an isometric projection (see Art. 631), viz., hy drawing chords of the 
circle, and tangents, then representing such auxiliary lines in oblique view and sketching the curve 
(now an ellipse) through the proper points. Fig. 408 illustrates this in full. 



OBLIQUE PROJECTION.— CAVALIER PERSPECTIVE. 



249 



647. Oblique projection is even better adaj^ted than isometric to the representation of timber fram- 
ings, machine and bridge details, aiid other objects in which ^igr- -^ca. 

straight lines — usually in mutually perpendicular directions — 
predominate, since all angles, curves, etc., lying in planes jxir- 
allel to the paper, appear of the same Jorm in projection, 
while the relations of lines perj^endicular to the paper are 
preserved by a simple ratio, ordinarilj' one of equality. 

648. "When the rays make with the plane of projection 
an angle greater than 45°, obliciue projections give etTects 
more closely analogous to a true perspective, since the fore- 
shortening is a closer apjsroximation to that ordinarily exi-st- 
ing from a finite point of view. This is illustrated by Fig. 
409, in which an object A B D E, known to be 1" thick, has its depth represented as only \" 
in the second view, instead of full size, as in a cavalier persj^ective, the front faces being the same 
size in each. Provided that the scale of reduction were known, ab cdkf would answer as well for 
a working drawing as a 45 "-projection. 

649. By way of contrast with an isometric view the timber framing represented by Fig. 398 is 




T'i.s- -ill- 









"s 


1 1 


i 


I " 




e \ f 






I ; 


\ 




B W A 



drawn in cavalier perspective in Fig. 410. Reference may advantageously be made, at this point, to 
Figs. 44, 45 and 46, which are oblique ^dews of one form. 

The keystone of the arch in Fig. 400, whose isometric view is shown in Fig. 401, appears in 
oblique projection in Fig. 411; the direction of lines not parallel to the axes of the circumscribing 
prism being found by "offsets" that mu.st be taken in Fig-- -iis- 

axial directions. 

650. Shadoics, in oblique jyrojection. As in other pro- 
jections, the conventional direction for the light is that Ar 
of the bodj'- diagonal of the oblique cube. The edges to 
draw in shade lines are obvious on inspection. (Fig. 412). 

651. An interesting application of oblique projection, 
earlier mentioned, is in the drawing of crystals. Fig. 
414 illustrates this, in the representation of a form common in fluorite and called the tetrahexahe- 
dron, bounded by twenty -four planes, each of which fulfills the condition expressed in the formula 




250 



THEORETICAL AND PRACTICAL GRAPHICS. 



00 : n : 1 ; that is, each face is parallel to one axis, cuts another at a unit's distance, and the third 

at some multiple of the unit. 

The three axes in this system are equal, and mutually per- 
pendicular; but their projected lengths are aa', bb', cc'. 

The direction of projectors which was assumed to give the 
lengths shown, was that of R N in Fig. 413, derived by turning the 
perpendicular CN through a horizontal angle CJVAT^ 18° 26', and 
then elevating it through a vertical angle MN^S^ 9° 28'; values 
that are given by Dana as well adapted to tlie exhibition of the 
forms occurring in this system. 

The axes once established, if we wish to construct on them the form oo : 2 : 1, we lay off on 





each (extended) one-half its own (projected) length; thus cc" and c' c'" each equal oc'; bb" equals 
ob, etc. Then draw in light lines the traces of the various faces on the planes of the axes. Thus, 
a'b" and a"b each represent the trace of a plane cutting the c-axis at infinity, and the other axes 
at either one or two units distance; the former intercepting the two units on the 6 -axis and the one 
on the a-axis, while for a"b it is exactly the reverse. Through the intersection of a'b" and a"b' 
a line is drawn parallel to the c-axis, indefinite in length at first, but determinate later by the 
intersection with it of other edges similarly found. 

The student may develop in the same manner the forms oo:3:l; oo:2:3; oo;3:4; oo:4:5. 



APPENDIX 




p. R. R. STANDARD RAIL SECTION. 

100 LBS. PER YARD. 



Draw the above either full size or enlarged 50 % . In either case 
draw section lines in Prussian blue, sjjacing not less than one -twentieth 
of an inch. Dimension lines, red. Dimensions and arrow heads, black. 
Lettering and numerals either in Extended Gothic or Reinhardt Gothic. 




ALLEN-RICHARDSON SLIDE VALVE. 

Draw either full size or larger. Section lines in Prussian blue, one -twentieth of an inch aj^art. 
Dimension lines, red. Dimensions and arrow heads, black. Lettering and numerals either in 
Extended Gothic or Reinhardt Gothic. 



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